On Finite Exceptional Orthogonal Polynomial Sequences Composed of Rational Darboux Transforms of Romanovski-Jacobi Polynomials

Abstract

The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential polynomial systems (X-Jacobi DPSs) of series J1, J2, J3, and W. The first three X-DPSs formed by pseudo-Wronskians of two Jacobi polynomials contain both exceptional orthogonal polynomial systems (X-Jacobi OPSs) on the interval (−1, +1) and the finite EOP sequences on the positive interval (1, ∞). On the contrary, the X-DPS of series W formed by Wronskians of two Jacobi polynomials contains only (infinitely many) finite EOP sequences on the interval (1, ∞). In addition, the paper rigorously examines the three isospectral families of the associated Liouville potentials (rationally extended hyperbolic Pöschl-Teller potentials of types a, b, and a′) exactly quantized by the EOPs in question.

1. Introduction

A decade ago the author [1] (under influence of Odake and Sasaki’s celebrated paper [2] and its extension in [3]) demonstrated the existence of the three isospectral families of the rational SUSY partners of the hyperbolic Pöschl-Teller (h-PT) potential [4] quantized by the rational Darboux transforms ($\mathrm{R}\mathrm{D}\mathfrak{T}\mathrm{s}$) of the Romanovski-Jacobi (R-Jacobi) polynomials [5,6,7]. In all the cases (labeled $\underset{˜}{\mathit{a}}$, $\underset{˜}{\mathit{b}}$, and ${\underset{˜}{\mathit{a}}}^{\prime }$ below for the reasons specified later) the quasi-rational transformation functions (q-RTFs) were represented by the principal Frobenius solutions (PFSs) of the Jacobi-reference (JRef) canonical Sturm-Liouville equation (CSLE) nonvanishing in the quantization interval (+1, ∞). The finite exceptional orthogonal polynomial EOP sequence reported in [3] was identified by us as type $\underset{˜}{\mathit{b}}$. As pointed to in [8], the infinitely many finite polynomial sequences of type ${\underset{˜}{\mathit{a}}}^{\prime }$ had been constructed by Grandati [9] a year earlier; however, he did not recognized the fact that, in contrast with other potentials discussed in the paper, the aforementioned polynomial sequences are X-orthogonal with a positive weight.

In the same year as [1], Yadav et al. [10] calculated the scattering amplitude for the rationally extended h-PT potential solvable by the finite sequence of the EOPs of type $\underset{˜}{\mathit{b}}$, referring to the latter simply as ‘X_m-Jacobi EOPs’. The epithet (repeatedly used by these authors in the more recent papers [11,12,13,14]) seemed confusing since the EOPs in question do not belong to the X_m-Jacobi orthogonal polynomial system (OPS) [15,16,17] and therefore cannot form its finite subset.

Before continuing our discussion, let us first point to the dubious use of the term ‘EOP’ in the literature, similar to the slang use of the term ‘orthogonal Jacobi polynomials’, instead of ‘classical Jacobi polynomials’, which disregards the existence of the finite orthogonal subsets formed by the R-Jacobi polynomials. Similarly, Gómez-Ullate, Milson et al. [15,16,18,19] use the term ‘EOPs’ as the synonym for ‘X-OPS’, disregarding the existence of the finite EOP sequences [1,2,3,8,10,11,12,13,20,21,22,23,24,25,26,27,28,29,30] represented by the rational Darboux transforms ($\mathrm{R}\mathrm{D}\mathfrak{T}\mathrm{s}$) of Romanovski polynomials [1,8,21,24,25,26,27,28,29]. (As a puzzling exception, Gómez-Ullate, Grandati, and Milson [31], when citing the studies on the EOPs, did mention the papers [9] and [20], which deal solely with the problems solved by the finite EOP sequences).

On the contrary, Yadav et al. [12], after giving credit to the three pioneering studies [15,32,33] on the X_m-Jacobi and X_m-Laguerre OPSs, made reference to a mixture of papers covering both infinite and finite EOP sequences. This statement (in contrast with their earlier remark [11] viewing the finite EOP sequences as subsets of the X-OPSs) is mostly consistent with our use of the term ‘EOP’.

The rigorous analysis of the aforementioned finite EOP sequences of types $\underset{˜}{\mathit{a}}$, $\underset{˜}{\mathit{b}}$, and ${\underset{˜}{\mathit{a}}}^{\prime }$ in [24] revealed that the EOP sequences of types $\underset{˜}{\mathit{a}}$ and $\underset{˜}{\mathit{b}}$ are formally generated by the same shift operators as the cases J1 and J2 case in [3], except that the indexes of the seed Jacobi polynomials in our case were independent of the polynomial degrees. This observation brought the author [24] to the concept of the exceptional differential polynomial systems (X-DPSs), with the term ‘DPS’ used in exactly the same sense as it was suggested by Everitt et al. [34,35] for conventional sequences of polynomials obeying the Bochner theorem [36]. In following the commonly accepted terminology suggested by Gómez-Ullate et al. [33] for the X₁-OPSs, we call the given DPS exceptional, because the polynomial sequence in question either does not start from a constant or lacks the first-degree polynomial and therefore do not obey the Bochner theorem. Indeed, as stressed by Kwon and Littlewood [37], Bochner himself “did not mention the orthogonality of the polynomial systems that he found. The problem of classifying all classical orthogonal polynomials was handled by many authors thereafter” based on his analysis of possible polynomial solutions of complex second-order differential eigenequations.

Compared with the rigorous mathematical analysis of the X-OPSs in [18,19], the concept of the X-DPSs put forward by us in [24] represents the parallel direction dealing with the solvable rational CSLEs (RCSLEs) and related X-Bochner ordinary differential equations (ODEs), instead of the (irregular) exceptional Bochner (X-Bochner) operators in [18] and related polynomial Sturm-Liouville problems (PSLPs) sketched in [15,18,32].

The interrelation between the two approaches is closely related to the dual use of the term ’Darboux transformation’ (DT), following the discovery by Andrianov et al. [38,39] that the renowned transformation of the Schrödinger equation initially suggested by Darboux [40] for the generic second-order canonical differential eigenequation (long before the birth of the quantum mechanics) is equivalent to its intertwining factorization. We refer the reader to the excellent overview of this issue in [41].

More recently Gómez-Ullate et al. [42] initiated the new direction in the theory of the rational Sturm-Liouville equations (RSLEs) by applying the intertwining factorization to the second-order differential eigenoperator. This operation was termed ’Darboux transformation’, based on the dualism existent in the particular case of the Schrödinger operator. This innovation followed by its extension to the X-OPSs [15,32,33] laid the foundation for their rigorous theory, which was advanced to the higher level in [18,19].

The author (being accustomed [43] to the strict use of the mentioned term) took the different turn [1] in the extension of the DTs to the SLEs. As we understand now, our original intuitive idea was based on the three-step operation:

(i)

the Liouville transformation from the CSLE to the Schrödinger equation;

(ii)

the Darboux deformation of the corresponding Liouville potential;

(iii)

the reverse Liouville transformation from the Schrödinger equation to the new CSLE using the same change of variable as at Step (i).

This was referred to by us years later [44] as ‘Liouville-Darboux transformation’. Note that we invented the term ‘Darboux deformation’ (DD), used instead of ‘DT’, simply to avoid the multiple repetition of the word ‘transformation’. It is worth stressing again that we give to this term its original meaning implied by Darboux [40].

The extensive exploration of the literature revealed that the transformation of the CSLE sketched above has been introduced by Rudyak and Zakhariev [45] in the late eighties. Schnizer and Leeb [46,47] named it ‘generalized Darboux transformation’ (GDT); this name was also accepted in some later studies on this subject. However, since various authors give to this widely used name completely different meanings in both physics and mathematics (see [44] for numerous examples), we suggested the aforementioned name ‘Liouville Darboux transformation’ as an alternative. Our current perception is that the latter is slightly misleading because it relates the definition of the transformation to the DD of the Liouville potential, which is absolutely irrelevant to the problem under consideration unless we are interested in quantum-mechanical applications.

Below we simply refer to the above operations as ‘Rudjak-Zakhariev transformations’ (RZTs) and consider its three-step decomposition suggested in [44] just as one of its realizations, but not as its definition (see Appendix A for more details). We term the RZT of the RCSLE as ‘rational RZT’ (RRZT) if it uses a q-RTF. The use of the RRZTs for constructing the RCSLEs with infinite sequences of quasi-rational solutions (q-RSs) constitutes the key element of the developed-by-us [8,24,25,26,27,28,29] theory of the X-DPSs formed by the polynomial components of the q-RSs of the given series.

In this paper we, based on the results of our previous studies [8,24], scrupulously analyze the manifold of the RCSLEs obtained by RRZTs of the Jacobi-reference (JRef) CSLE, which is defined via (1)–(3) in Section 2. Each q-RS formed by a m-degree Jacobi polynomial can be used as the q-RTF giving rise to the RCSLE with m + 2 poles in the finite complex plane. The Jacobi indexes of the seed polynomials are defined in the segments carved by the three vertical and three horizontal lines with the abscissas and respectively ordinates equal to −1, 0, and +1.

Each transformed CSLE is then converted to the Bochner-type ODE with polynomial coefficients, taking advantage of the fact that the density function of our interest has only simple poles in the finite plane and as a results the mentioned gauge transformation is energy-independent [1]. Consequently, the linear coefficient function of the resultant differential equation does not depend on the degrees of the sought-for polynomials.

It has been proven in [24] that each transformed RCSLE constructed in such a way has a quartet of infinite sequences of q-RSs with polynomial components forming the four X_m-Jacobi DPSs of series J1, J2, D, and W, as they were termed by us. The shift operators for the cases J1 and J2 in [3] match our equations for the X_m-Jacobi DPSs of series J1 and J2 accordingly, though, as pointed to in [8], the indexes of the seed Jacobi polynomials in our scheme are independent of the polynomials, in a sharp contrast with [3]. The innovative concept of the X_m-Jacobi DPSs containing both infinite and finite EOP sequences constitutes the cornerstone of our formalism.

Another fundamentally important part of our approach is the formulation of the rational Sturm-Liouville problem (RSLP) to detect infinite or finite orthogonal subsequences in each of the four X_m-Jacobi DPSs. Note that, until now, we have not mentioned any restriction on the zeros of the seed Jacobi polynomials. However, to select the RSLPs solvable by either infinite or finite EOP sequences, we have to focus solely on the seed Jacobi polynomials with no zeros inside the quantization interval. At this point we come to the main advantage of our approach, compared with the general theory of X-OPSs advanced by Garcia-Ferrero et al. [18,19]. Namely we formulate the RSLP for both finite and infinite quantization intervals. In contrast, the X_m-Jacobi OPSs appear only if the OBCs for the given SLP are imposed at the ends of the finite interval (−1, +1). Since the cited authors were interested only in the X-OPSs they term the X-Bochner operator ‘regular’, if the q-RTF used to generate the RCSLE in question does not have nodes inside the interval (−1, +1). On the contrary, we have to additionally specify the open interval, where the RCSLE of our interest may not have any singularities. For the purposes of this paper (unless explicitly specified otherwise) the q-RTF is termed regular if does not have real nodes larger than 1. Similarly we refer to the RRZT as regular (reg-RRZT) if the corresponding transform of the JRef CSLE does not have poles in the interval (1, ∞).

A certain deficiency of our Sturm-Liouville approach, compared with the intertwining technique advanced in [18,19], is that we [48] require that the RRZT of the RSLE preserves both the leading and weight coefficient functions. As a result, the RRZTs, as they define here, represent only a narrow subset of the rational Darboux transformations (RDTs) in the terminology of Garcia-Ferrero et al. [18,19].

As it has been already done above in our references to the $\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}\mathrm{s}$ of the Romanovsky polynomials, we will often use the commonly accepted term ‘$\mathrm{R}\mathrm{D}\mathfrak{T}$’, instead of ‘$\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}$’, despite the fact that we have to introduce the more restrictive requirement for both indexes of the classical Jacobi polynomial or the first index of the R-Jacobi polynomial to be positive.

Our next step is to find all quasi-rational solutions (q-RSs) of the JRef CSLE, which do not have nodes inside the selected quantization interval, which assures that the transformed RCSLE does not have poles inside the interval of our interest. It was taken for granted in our earlier works [1,8,24] that any PFS below the lowest eigenvalue is necessarily nodeless. As one can see from the proof presented in Appendix B, this is not by any means a trivial (though wide-spread) presumption.

Using Klein’s formulas (see §6.72 in [49]) for the numbers of zeros of a Jacobi polynomial in the intervals (−∞, −1), (−1, +1), and (+1, ∞), we showed [1] that the JRef CSLE also has irregular-at-both-ends q-RSs (type $\tilde{\mathit{d}}$) with no nodes in the interval (+1, ∞). Keeping in mind the RRZT using this q-RS as q-RTF inserts the extra energy level below the lowest eigenvalue, it can be called [41] ‘dressing’ transformation.

To find the orthogonal $\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}\mathrm{s}$ of the classical and R-Jacobi polynomials, we convert the transformed RCSLE to its prime form defined in such a way [48] that the Dirichlet boundary condition (DBC) unambiguously selects the PFS near the singular end in question. As a result, the DBCs at the ends of the given quantization interval select the solutions representing the PFSs near both singular ends. This is the brand new approach, which has allowed us to treat both infinite and finite EOP sequences in parallel, in contrast with the conventional technique [15,16,17] paying attention solely to the X-OPSs. The technique advanced in our papers [8,24,25,26,27] made it possible to develop the uniform approach to the $\mathrm{R}\mathrm{D}\mathrm{C}\mathfrak{T}\mathrm{s}$ of the three families of the Romanovski polynomials, with the current paper focusing exclusively on the $\mathrm{R}\mathrm{D}\mathfrak{T}\mathrm{s}$ of the R-Jacobi polynomials.

In the limit point (LP) region the Dirichlet problem mentioned above unequivocally specifies the eigenfunctions square integrable with the weight function of the SLE under consideration. Our procedure also provides the prescription for constructing at least some of the EOP sequence existent in the limit circle (LC) region, where the corresponding Liouville potential has only continuously degenerate bound energy states (CDBESs).

We found that the X_m-Jacobi DPSs of both series J1 and J2 contain X_m-Jacobi OPSs, which are interrelated via the reflection of their argument accompanied by the interchange of the Jacobi indexes. For this reason Gómez-Ullate et al. [15,16] restricted their analysis to the properties of one of them (which happened to be the X_m-Jacobi OPS of series J2 in our terms), referring to the latter simply as ‘X_m-Jacobi polynomials’ and dropping the X_m-Jacobi OPS of series J1 from any future consideration. To be consistent with the commonly adopted terminology in [15,16], we will term polynomials from the X_m-Jacobi DPS of series J2 and the X_m-Jacobi OPS of series J2 as ‘X_m-Jacobi polynomials’ and ‘orthogonal X_m-Jacobi polynomials’ accordingly, by analogy with the conventional terms ‘Jacobi polynomials’ and ‘orthogonal (classical) Jacobi polynomials’. Skipping the epithet ‘orthogonal’ in [15,16] can be thus treated as a slang term, similar to the widespread practice of omitting it in references to orthogonal Jacobi polynomials.

On other hand, our analysis revealed that the infinitely many finite EOP sequences of type $\underset{˜}{\mathit{a}}$ and limitedly many the finite EOP sequences of type $\underset{˜}{\mathit{b}}$ (constructed using the PFSs near the origin and at infinity respectively) belong to the X_m-Jacobi DPS of series J1 and accordingly J2, which makes it necessary to analyze both X-DPSs in parallel.

In particular, as discussed in Section 4.2 below, the X_m-Jacobi DPS of series J2 contains the X_m-Jacobi OPS (in terms of [15,16]), the finite EOP sequence orthogonal on the negative interval (−∞, −1), and another finite EOP sequence orthogonal on the positive interval (1, ∞). The latter EOP sequence of type $\tilde{\mathit{b}}$ is composed of the $\mathrm{R}\mathrm{D}\mathfrak{T}\mathrm{s}$ of the R-Jacobi polynomials, which represent the polynomial components of the eigenfunctions [10,11,12] of the rationally extended h-PT potential [11,12].

Regretfully, both our works [8] and [24] overlooked the important modification in the definition of the ‘X_m-Jacobi polynomials’ by Yadav et al. in [11]. Namely the indexes α and β appearing in the expression for X_m-Jacobi polynomials’ in [10] were defined independently of the degree of the seed Jacobi polynomials and thereby (as illuminated in Section 4.2 below) made the mentioned expression fully consistent with our definition of the X_m-Jacobi DPS of series J2.

As for the X_m-Jacobi DPS of series J1, it contains the OPS (‘X_m-Jacobi OPS of series J1 in our terms), the finite EOP sequence orthogonal on the negative interval (−∞, −1), and another finite EOP sequence (this time of type $\underset{˜}{\mathit{a}}$) orthogonal on the positive interval (1, ∞). Obviously, the finite EOP sequence orthogonal on the negative interval (−∞, −1) can be obtained from the EOP sequence of type $\underset{˜}{\mathit{b}}$ by the reflection of the argument, followed by the interchange of the indexes; i.e., the two finite EOP sequences in question are interrelated in exactly the same way as the DPSs of series J1 and J2 to which they belong.

The third EOP sequence of type ${\underset{˜}{\mathit{a}}}^{\prime }$ generated using the PFSs near origin (similarly to EOP sequence of type $\underset{˜}{\mathit{a}}$) constitutes the orthogonal subset of the X_m-Jacobi DPS of series W composed of the Wronskians of two Jacobi polynomials with common pairs of the indexes. As it has been already pointed to by us in [8], the Wronskian transforms of the R-Jacobi polynomials have been already brought to light in the cited article [9] by Grandati. However, he did not realized that the constructed polynomial Wronskians are X-orthogonal. Indeed, the rational realization of the h-PT potential (using the variable cosh 2x) belongs to group A in Odake and Sasaki’s classification scheme [22] of the rational translationally shape-invariant (RTSI) potentials (contrary to the other RTSI potentials discussed in [9], which all belong to Group B). As a result, any admissible RDT of the R-Jacobi polynomials results in a finite EOP sequence.

Ironically the most obvious rational extension of the h-PT potential, using the TS with the polynomial component represented by the classical Jacobi polynomials and therefore necessarily nodeless on the interval (1, ∞), has been never discussed in the literature (apart from our works). The very remarkable feature of the finite EOP sequences of type $\underset{˜}{\mathit{a}}$ discovered in [24] is that they can be arranged into the rectilinear polynomial matrix with a finite number of rows and an infinite number of columns composed of the X-Jacobi orthogonal polynomial system (X-Jacobi OPS) and $\mathrm{R}\mathrm{D}\mathfrak{T}\mathrm{s}$ of the R-Jacobi polynomials.

Finally, the EOP sequences of type $\tilde{\mathit{d}}$ belong to the X_m-Jacobi DPS of series D. We introduced this label in [24] to stress that the DPS in question is composed of the so-called [48] ‘polynomial determinants’ (PDs). However, after realizing [8] that this X-DPS contains the X_m-Jacobi OPS of series J3 discovered by Grandati and Bérard [50] we switched to the term ‘X_m-Jacobi DPS of series J3′, i.e., the X_m-Jacobi OPSs of series j1, j2, and J3 constitute infinite orthogonal subsets of the X_m-Jacobi DPSs of series J1, J2, and J3 respectively.

To summarize, let us note that, compared with the PSLPs roughed out in [15,18,33], the DBCs at the ends of the quantization interval [−1,+1] cover only the rational Rudjak-Zakhariev transforms ($\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}\mathrm{s}$) of the classical Jacobi polynomials with positive indexes as it has been already demonstrated in [29] for m = 1. On the other hand, our technique is sufficient to find all the rational Liouville potentials solvable in terms of either infinite and finite X_m-Jacobi EOP sequences, making it possible to construct the new finite EOP sequences.

One of the most important achievements of this paper (in addition to the systematic description and more precise summary of the earlier results spread between the three preprints [1,8,24]) is the representation of the X_m-Jacobi DPSs of series J1, J2, and J3 (and therefore the corresponding X_m-Jacobi OPSs) via the ‘pseudo-Wronskian polynomials’ ($\mathrm{p}\text{-}\mathrm{W}\mathrm{P}\mathrm{s}$) of two Jacobi polynomials [51]. As outlined in Section 6, this fresh development opens a promising new direction in the theory of both infinite and finite EOP sequences composed of ‘simple’ $\mathrm{p}\text{-}\mathrm{W}\mathrm{s}$ of several seed Jacobi polynomials with common indexes and a single either classical Jacobi or R-Jacobi polynomial.

2. Quantization of JRef CSLE on Infinite Interval (1, ∞)

Let us start our analysis with the Jacobi-reference (JRef) CSLE

$$\left\{{\displaystyle \frac{{\mathrm{d}}^2 }{\mathrm{d}{\mathsf{\eta }}^2}}+{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]+\underset{˜}{\mathsf{\epsilon }} \mathit{\rho } [\mathsf{\eta }] \right\} \mathsf{\Phi } [\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}; \underset{˜}{\mathsf{\epsilon }}]= 0 (\mathsf{\eta }>1)$$

(1)

with the single pole density function

$$\mathit{\rho } [\mathsf{\eta }]:= {\displaystyle \frac{1}{{\mathsf{\eta }}^2-1}}>0$$

(2)

positive on the infinite interval (1, ∞). The reference polynomial fraction (RefPF) is parameterized as follows:

$${\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}] := {\displaystyle \sum _{\aleph =\pm }}{\displaystyle \frac{1-{\mathsf{\lambda }}_{\mathrm{o};\aleph }^2}{4{(1-\aleph \mathsf{\eta })}^2}}+ {\displaystyle \frac{1-{\mathsf{\lambda }}_{\mathrm{o};+}^2-{\mathsf{\lambda }}_{\mathrm{o};-}^2}{4(1-{\mathsf{\eta }}^2)}}$$

(3)

$$=-{\displaystyle \frac{1}{2({\mathsf{\eta }}^2-1)}} {\displaystyle \sum _{\aleph =\pm }}{\displaystyle \frac{1-{\mathsf{\lambda }}_{\mathrm{o};\aleph }^2}{1-\aleph \mathsf{\eta }}}+{\displaystyle \frac{1}{4({\mathsf{\eta }}^2-1)}},$$

(4)

where ${\mathsf{\lambda }}_{\mathrm{o};\pm }$ are the exponent differences (ExpDiffs) for the poles at ±1 and the energy reference point is chosen via the requirement that the ExpDiff for the singular point at infinity vanishes at zero energy, i.e.,

$$\underset{|\mathsf{\eta }|\to \infty }{lim}\left({\mathsf{\eta }}^2{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\right)={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}.$$

(5)

In following [29], we underline ε by tilde to distinguish it from the spectral parameter $\mathsf{\epsilon }=-\underset{˜}{\mathsf{\epsilon }}$ for the JRef CSLE defined on the conventional interval (−1, +1):

$$\left\{{\displaystyle \frac{{\mathrm{d}}^2 }{\mathrm{d}{\mathsf{\eta }}^2}}+{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]+\mathsf{\epsilon } \mathit{\rho } [\mathsf{\eta }] \right\} \mathsf{\Phi } [\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}; \mathsf{\epsilon }]= 0(-1<\mathsf{\eta }<1),$$

(6)

with the density function (2) changed for

$$\mathit{\rho } [\mathsf{\eta }]:= {\displaystyle \frac{1}{1-{\mathsf{\eta }}^2}}$$

(7)

2.1. Liouville Transformation of JRef CSLE on Infinite Interval

The gauge transformation

$$\mathsf{\Psi } [\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}; \underset{˜}{\mathsf{\epsilon }}]= {\mathit{\rho }}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}} [\mathsf{\eta }] \mathsf{\Phi } [\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}; \underset{˜}{\mathsf{\epsilon }}]$$

(8)

converts the JRef CSLE (1) into the ‘algebraic’ [48] Schrödinger equation

$$\left\{ {\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}\sqrt{{\mathsf{\eta }}^2-1}{\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}+\left(\underset{˜}{\mathsf{\epsilon }}-\mathrm{V}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\right)\sqrt{{\mathsf{\eta }}^2-1}\right\} \mathsf{\Psi } [\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}; \underset{˜}{\mathsf{\epsilon }}]= 0$$

(9)

with the Liouville potential

$$\mathrm{V}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=-({\mathsf{\eta }}^2-1) {\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}Schw\{\sqrt{{\mathsf{\eta }}^2-1} \},$$

(10)

where the ‘Schwarzian’ operator

$$Schw\{\mathrm{f}[\mathsf{\eta }]\}=\mathrm{f}[\mathsf{\eta }]\stackrel{••}{\mathrm{f}}[\mathsf{\eta }]-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\stackrel{•}{\mathrm{f}}}^2[\mathsf{\eta }]$$

(11)

(with the dot standing for the derivatives with respect to η) turns into the conventional Schwarzian derivative $\{\mathsf{\eta },\mathrm{r}\}$ [52] if the change of variable $\mathsf{\eta }(\mathrm{r})$ satisfies the first-order ODE

$${\mathsf{\eta }}^{\prime }(\mathrm{x})={\mathrm{f}}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }(\mathrm{x})],$$

(12)

with the prime symbol standing for the derivative with respect to x. Note that the companion Liouville potential obtained by the Liouville transformation on the finite interval (−1, +1) can be obtained from (8) simply by replacing ${\mathsf{\eta }}^2-1$ for its absolute value which changes sign of the first term in (10) while keeping unchanged the second. This remarkable feature of the JRef CSLE with the density function (7) covering both versions of the PT potential [4] forms the basis for the unified approach to the rational extensions of these potentials recently suggested in [30].

Substituting (4) and

$$Schw\{\sqrt{{\mathsf{\eta }}^2-1}\}=-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+{\displaystyle \frac{3}{4(1-\mathsf{\eta })}}+{\displaystyle \frac{3}{4(1+\mathsf{\eta })}}$$

(13)

into (10) gives

$$\mathrm{V}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]={\displaystyle \frac{{\mathsf{\lambda }}_{\mathrm{o};+}^2-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{2(\mathsf{\eta }-1)}}-{\displaystyle \frac{{\mathsf{\lambda }}_{\mathrm{o};-}^2-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{2(\mathsf{\eta }+1)}}$$

(14)

It is essential that the potential function (14) vanishes at infinity due to our choice of the energy reference point.

The change of variable

$$\mathsf{\eta }(\mathrm{x})=cosh \mathrm{x}$$

(15)

finally brings us to the h-PT potential in its conventional Pöschl -Teller form [4]

$${\mathrm{V}}_{\mathrm{h}-\mathrm{P}\mathrm{T}}(\mathrm{r};{\mathsf{\lambda }}_{\mathrm{o};+},{\mathsf{\lambda }}_{\mathrm{o};-}]={\displaystyle \frac{{\mathsf{\lambda }}_{\mathsf{o};+}^2-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{4sin{h}^2\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}- {\displaystyle \frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-{\mathsf{\lambda }}_{\mathsf{o};-}^2}{4cos{h}^2\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} (0<\mathrm{x}<\infty ),$$

(16)

where the energy-independent ExpDiffs ${\mathsf{\lambda }}_{\mathrm{o};+}$ and ${\mathsf{\lambda }}_{\mathrm{o};-}$ at the finite singular points are related to Odake and Sasaki’s parameter g and h in [2] as follows

$$\mathrm{g}={\mathsf{\lambda }}_{\mathrm{o};+}+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}, \mathrm{h}={\mathsf{\lambda }}_{\mathrm{o};-}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}.$$

(17)

It is worth noting that that the first of the listed parameters coincides with the larger characteristic exponent (ChExp) for the pole of the given radial Schrödinger equation at the origin, i.e., the potential is repulsive iff $\mathrm{g}>\mathrm{1}$(${\mathsf{\lambda }}_{\mathrm{o};+}>{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$). The constraint $\mathrm{g}>{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ (${\mathsf{\lambda }}_{\mathrm{o};+}>\mathrm{1}$) specifies the necessary and sufficient condition for the radial potential to have the discrete energy spectrum, whereas any solution of the radial Schrödinger equation is squarely normalizable for $\mathrm{1}<\mathrm{g}<{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ (${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}<{\mathsf{\lambda }}_{\mathrm{o};+}<1$).

To be able to compare our results with those in [11,12], let us also consider the alternative parametrization of the h-PT potential introduced in the renown review article by Cooper et al. [53] and then adopted by Bagchi et al. [20] in their exhaustive analysis of the m = 1 rational extension of the h-PT potential. Setting

$$2\mathrm{A}+1:={\mathsf{\lambda }}_{\mathrm{o};-}-{\mathsf{\lambda }}_{\mathrm{o};+},2\mathrm{B}={\mathsf{\lambda }}_{\mathrm{o};-}+{\mathsf{\lambda }}_{\mathrm{o};+}$$

(18)

representing the sum of the PFs in the right-hand side of (13) as

$$\mathrm{V}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]={\mathrm{V}}_{\mathrm{A},\mathrm{B}}[\mathsf{\eta }]:={\displaystyle \frac{\mathrm{A}(\mathrm{A}+1)+{\mathrm{B}}^2-(2\mathrm{A}+1)\mathrm{B}\mathsf{\eta } }{{\mathsf{\eta }}^2-1}}$$

(19)

and expressing the latter in terms of x ≡ r via (15) brings us to the potential function (1) in [11]. By definition,

$$\mathrm{B}-\mathrm{A}={\mathsf{\lambda }}_{\mathrm{o};+}+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\mathrm{g}>{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}, \mathrm{B}+\mathrm{A}={\mathsf{\lambda }}_{\mathrm{o};-}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\mathrm{h}>-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},$$

(20)

i.e., $\mathrm{B}>\mathrm{A}+$¹/₂ The given potential is repulsive iff ${\mathsf{\lambda }}_{\mathrm{o};+}>$¹/₂, i.e., $\mathrm{B}>\mathrm{A}+1$ [20], with no limitation on either sign or a value of the parameter A. The additional requirement for the parameter A to be positive [20] only assures that the potential has at least one bound energy level.

2.2. Quartet of q-RSs

One can directly verify that the quasi-rational function

$${\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]:={(1+\mathsf{\eta })}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{-}+1)}{(\mathsf{\eta }-1)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{+}+1)},$$

(21)

is the solution of the JRef CSLE (6) at the energy $\mathsf{\epsilon }={\mathsf{\epsilon }}_0(\stackrel{\rightharpoonup }{\mathsf{\lambda }}),$ or alternatively the solution of the JRef CSLE (1) at energy $\underset{˜}{\mathsf{\epsilon }}=-{\mathsf{\epsilon }}_0(\stackrel{\rightharpoonup }{\mathsf{\lambda }})$,

$${\mathsf{\epsilon }}_0(\stackrel{\rightharpoonup }{\mathsf{\lambda }})={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{({\mathsf{\lambda }}_{+}+{\mathsf{\lambda }}_{-}+1)}^2$$

(22)

Substituting the quasi-rational function

$${\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]={\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })(|{\mathsf{\lambda }}_{\pm }|={\mathsf{\lambda }}_{\mathrm{o};\pm })$$

(23)

into the JRef CSLE (6) and taking into account that the quasi-rational function (21) satisfies the first-order differential equation:

$${\stackrel{•}{\mathsf{\varphi }}}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]={\displaystyle \frac{P_1^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })}{{\mathsf{\eta }}^2-1}}{\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}],$$

(24)

with dot standing for the derivative with respect to η, we find that (23) is the solutions of the given equation at the energy

$${\mathsf{\epsilon }}_{\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})\equiv -{\underset{˜}{\mathsf{\epsilon }}}_{\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}) :={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{({\mathsf{\lambda }}_{+}+{\mathsf{\lambda }}_{-}+2\mathrm{m}+1)}^2,$$

(25)

provided its polynomial component satisfies the Jacobi equation:

$$\begin{gathered} ({\mathsf{\eta }}^2-1){\stackrel{•}{\mathrm{P}}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })+2{\mathrm{P}}_1^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta }) {\stackrel{•}{\mathrm{P}}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta }) \\ \left[{\mathsf{\epsilon }}_0(\stackrel{\rightharpoonup }{\mathsf{\lambda }})-{\mathsf{\epsilon }}_{\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})\right] {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })=0, \end{gathered}$$

(26)

The vital point is that the coefficient function for the first derivative of the Jacobi polynomial is independent of the polynomial degree, in the sharp contrast with the general case of the JRef CSLE solvable by Jacobi polynomials with degree-dependent indexes [54,55]. This is the intrinsic feature of the translationally form-invariant (TFI) CSLEs of group A, and in particular, from all the RTSI potentials discussed in [9], the stated assertion holds only for the h-PT potential.

One can easily verify that the shift parameters λ in Odake-Sasaki’s recipe [2] for constructing the rational SUSY partners of the t- and h-PT potentials are nothing but the indexes of the Jacobi polynomials forming the quasi-rational eigenfunctions of the mentioned potentials. Since the JRef PF (4) depends on the squares of the Jacobi indexes, it can be re-expressed in terms of the latter parameters, instead of the ExpDiffs ${\mathsf{\lambda }}_{\mathrm{o};\pm }$. This duplicate parametrization of the t- and h-PT potentials by the Jacobi indexes represents the basis for the aforementioned unified approach [30] to the $\mathrm{R}\mathrm{D}\mathfrak{T}\mathrm{s}$ of these potentials using the first-degree (m = 1) seed polynomials.

Note that Gómez-Ullate et al. [51] also parametrized the t-PT potential by means of the mentioned Jacobi indexes, except that the zero-point energy was randomly shifted by a constant dependent on the index signs. Examination of the JRef PF (4) reveals that the parameters α and β in (37) in [51] are allowed to take only positive values other than 1. In the case of ${\mathsf{\lambda }}_{\mathrm{o};+}=1$ (or ${\mathsf{\lambda }}_{\mathrm{o};-}=1$) the corresponding second-order pole disappears and the resultant problem with only the simple pole at +1 (or respectively at −1) requires a special attention). It will be shown in Section 5.1 below that one can formulate the Dirichlet problem for any positive values of ${\mathsf{\lambda }}_{\mathrm{o};\pm }\ne 1$, with no need to exclude the intervals $0<{\mathsf{\lambda }}_{\mathrm{o};\pm }\le$ ½ (in contrast with the constraint imposed on the Jacobi indexes α and β in [51]).

From a more general perspective, the absolute values of the shift parameters λ coincide with the ExpDiffs for the persistent poles of the transformed rational CSLEs in the finite plane. The latter interpretation of the shift parameters introduced in [2] allowed the cited authors to extend their approach to the rational Darboux-Crum transformations (RDCTs) of the t- and h-PT potentials [22]. We shall come back to the discussion of this issue in Section 6.

Restricting the Jacobi indexes by the condition:

$${\mathsf{\lambda }}_{-}, {\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-}+{\mathsf{\lambda }}_{+}+\mathrm{m}\ne -\mathrm{k}$$

(27)

for any positive integer $\mathrm{k}\le \mathrm{m}$ (see §4.22(3) in [49]; cf. (102) in [15] or (88) in [16]), we assure that the leading coefficient of the Jacobi polynomial of degree m,

$$\underset{\mathsf{\eta }\to \infty }{lim}{\mathsf{\eta }}^{-\mathrm{m}} {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })=2^{-\mathrm{m}}{〈\mathrm{m}+{\mathsf{\lambda }}_{+}+{\mathsf{\lambda }}_{-}+1〉}_{\mathrm{m}}/\mathrm{m}!$$

(28)

differs from zero and therefore the Jacobi polynomial in question has exactly m simple zeros ${\mathsf{\eta }}_l(\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathrm{m}).$ It is crucial that the Jacobi indexes do not depend on the polynomial degree, in contrast with the general case [54,55]. This remarkable feature of the CSLE under consideration is the direct consequence of the fact that the density function (2) has only simple poles in the finite plane [25] and as a result the ExpDiffs for the CSLE poles at ±1 become energy independent [1].

2.3. Prime SLE on Infinite Interval (1, ∞)

Instead of the conventional prerequisite of the spectral theory requiring the eigenfunctions to be normalizable with the weight $\mathit{\rho } [\mathsf{\eta }]$ we require that the boundary conditions of our choice unambiguously select the PFS near each singular endpoint. To achieve this goal, we first convert the given CSLE to its ‘prime’ form (p-SLE) selected by the requirement that the ChExps of the two Frobenius solutions near the given end differ only by their sign and as a result the DBC pinpoints the PFS.

The rational prime form of the JRef CSLE on the interval (1, ∞) is obtained by requiring the leading coefficient to be the first-degree polynomial

$$\cancel{p}[\mathsf{\eta }]=\mathsf{\eta }-1 \mathrm{f}\mathrm{o}\mathrm{r} \mathsf{\eta }\in [1,\infty ),$$

(29)

which brings us to the p-SLE [8,29]

$$\{-{\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}(\mathsf{\eta }-1) {\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}+\cancel{q}[\mathsf{\eta }; {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]+\underset{˜}{\mathsf{\epsilon }}\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}] \} \underset{˜}{\cancel{\mathsf{\Psi }}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\underset{˜}{\mathsf{\epsilon }}]= 0 (\mathsf{\eta }>1)$$

(30)

with the zero-energy free term

$$\cancel{q}[\mathsf{\eta }; {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=-(\mathsf{\eta }-1){\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta }; {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]+{\displaystyle \frac{1}{4(\mathsf{\eta }-1)}} (\mathsf{\eta } >1)$$

(31)

and the positive weight

$$\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]={(1+\mathsf{\eta })}^{-1}<{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2 $}\right.} (\mathsf{\eta }>1) .$$

(32)

Taking into account that

$$\underset{\mathsf{\eta }\to 1}{lim}\left[(\mathsf{\eta }-1)\cancel{q}[\mathsf{\eta }; {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\right]={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\mathsf{\lambda }}_{\mathrm{o};+}^2 >0$$

(33)

we confirm that the ChExps of the Frobenius solutions for the pole at +1 have the same absolute value ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathsf{\lambda }}_{\mathrm{o};+}$ while differing by their sign. The ExpDiff for the pole of the JRef CSLE (1) at infinity turned out to be energy dependent. Combining (30) with (4), one can verify that

$$\underset{\mathsf{\eta }\to \infty }{lim}\left[\mathsf{\eta }\cancel{q}[\mathsf{\eta }; {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\right]=-\underset{\mathsf{\eta }\to \infty }{lim}\left[{\mathsf{\eta }}^2{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\right]+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}=0,$$

(34)

confirming that the ChExps of the Frobenius solutions for the pole at ∞ are real only at negative energies and have in this case the same non-zero absolute value ¹/₂$\sqrt{-\mathsf{\epsilon }}$ while differing by their sign.

Combining (34) with the similar limit

$$\underset{\mathsf{\eta }\to 1+}{lim}\left[(\mathsf{\eta }-1)\cancel{q} [\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}\right]={\mathsf{\lambda }}_{\mathrm{o};\pm }^2 >0$$

(35)

for the pole at +1, we conclude that the PFSs of the p-SLE (30) near both singular endpoints are unambiguously determined by the Dirichlet boundary conditions (DBCs):

$$\underset{\mathsf{\eta }\to r}{lim}{\underset{˜}{\cancel{\mathsf{\Psi }}}}_r[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\underset{˜}{\mathsf{\epsilon }}]=0 (r=1,\infty )$$

(36)

and the sought-for eigenfunctions simply turn into the solutions of the p-SLE

$$\{-{\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}(\mathsf{\eta }-1) {\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}+\cancel{q}[\mathsf{\eta }; {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]+{\underset{˜}{\mathsf{\epsilon }}}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}] \} {\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]= 0$$

(37)

solved under the DBCs (36).

For ${\mathsf{\lambda }}_{\mathrm{o};+}>1$ the DBCs (35) unambiguously specify all the possible squarely integrable solutions of the p-SLE (31). On the other hand, any solution of this SLE is squarely integrable for $0<{\mathsf{\lambda }}_{\mathrm{o};+}<1$ which implies that the CSLE in question and consequently the corresponding Liouville potential (16) has the CDBESs. The OBC at η = 1 simply selects the squarely integrable solution representing the PFS near this singular end. It will be demonstrated in Section 5 that the $\mathrm{R}\mathrm{D}\mathfrak{T}\mathrm{s}$ of the latter solution play the important role in constructing the finite EOP sequences in the LC region ($0<{\mathsf{\lambda }}_{\mathrm{o};+}<1$).

2.4. R-Jacobi Polynomials

One can directly verify that the q-RSs

$${\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };-{\mathsf{\lambda }}_{\mathrm{o};-},{\mathsf{\lambda }}_{\mathrm{o};+}]:={\displaystyle \frac{1}{\sqrt{\mathsf{\eta }-1}}}{\mathsf{\varphi }}_{- +,\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}](1\le \mathsf{\eta }<\infty )$$

(38)

$$={\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };-{\mathsf{\lambda }}_{\mathrm{o};-},{\mathsf{\lambda }}_{\mathrm{o};+}]{\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{\mathrm{o};+},-{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta }),$$

(39)

at the energies

$${\underset{˜}{\mathsf{\epsilon }}}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}) ={\mathsf{\epsilon }}_{-+,\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}) :=-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{({\mathsf{\lambda }}_{\mathrm{o};-}-{\mathsf{\lambda }}_{\mathrm{o};+}-2\mathrm{j}-\mathrm{1})}^2,$$

(40)

with

$${\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]={(1+\mathsf{\eta })}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}(1+{\mathsf{\lambda }}_{-}^{\prime })} {(1-\mathsf{\eta })}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathsf{\lambda }}_{+}^{\prime }} (1\le \mathsf{\eta }<\infty ),$$

(41)

satisfy the DBCs (35) for

$$0\le \mathrm{j}\le {\mathrm{j}}_{\max }\equiv ⌊{\mathsf{\lambda }}_{\mathrm{o};-}-{\mathsf{\lambda }}_{\mathrm{o};+}-1⌋$$

(42)

and therefore represent the eigenfunctions of the p-SLE (36), normalizable with the weight (31):

$${\displaystyle \underset{1}{\overset{\infty }{\int }}\mathrm{d}\mathsf{\eta }} {\cancel{\mathsf{\psi }}}_{\mathrm{j}}^2[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]<\infty . (0\le \mathrm{j}\le {\mathrm{j}}_{\max }).$$

(43)

Since these are the eigensolutions of the Sturm-Liouville problem solved under the DBCs they must be also mutually orthogonal [56] with the weight (31):

$${\displaystyle \underset{1}{\overset{\infty }{\int }}\mathrm{d}\mathsf{\eta }} {\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]{\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=0 \mathrm{f}\mathrm{o}\mathrm{r} 0\le {\mathrm{j}}^{\prime }<\mathrm{j}\le {\mathrm{j}}_{\max },$$

(44)

which brings us to the conventional orthogonality relations for the R-Jacobi polynomials

$${\displaystyle \underset{0}{\overset{\infty }{\int }}\mathrm{d}\underrightarrow{\mathrm{z}}} {\underrightarrow{\mathsf{\omega }}}_{\mathsf{\alpha },\mathsf{\beta }}[\underrightarrow{\mathrm{z}}] {\mathrm{J}}_{\mathrm{j}}^{(\mathsf{\alpha },\mathsf{\beta })}(\underrightarrow{\mathrm{z}}){\mathrm{J}}_{{\mathrm{j}}^{\prime }}^{(\mathsf{\alpha },\mathsf{\beta })}(\underrightarrow{\mathrm{z}})=0 (\mathrm{j}\ne {\mathrm{j}}^{\prime })$$

(45)

with the weight

$${\underrightarrow{\mathsf{\omega }}}_{\mathsf{\alpha },\mathsf{\beta }}[\underrightarrow{\mathrm{z}}]:={\underrightarrow{\mathrm{z}}}^{\mathsf{\alpha }} {(\underrightarrow{\mathrm{z}}+1)}^{-|\mathsf{\beta }|} \mathrm{f}\mathrm{o}\mathrm{r} \underrightarrow{\mathrm{z}}\in [1,\infty )$$

(46)

under the constraint:

$$\mathsf{\alpha }:={\mathsf{\lambda }}_{\mathrm{o};+}\equiv \mathrm{B}-\mathrm{A}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}>0, \mathsf{\beta }:=-{\mathsf{\lambda }}_{\mathrm{o};-}\equiv -\mathrm{A}-\mathrm{B}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}<0.$$

(47)

Here we adopted Askey’s [57] definition of the R-Jacobi polynomials which, as proven by Chen and Srivastava [58], is equivalent to the elementary formula

$${\mathrm{J}}_{\mathrm{n}}^{(\mathsf{\alpha }, \mathsf{\beta })} (\underrightarrow{\mathrm{z}}) := {\mathrm{P}}_{\mathrm{n}}^{( \mathsf{\alpha },\mathsf{\beta })} (2\underrightarrow{\mathrm{z}}+1) \mathrm{f}\mathrm{o}\mathrm{r} \mathsf{\alpha }>-1, \mathsf{\beta }<0,$$

(48)

with

$$\underrightarrow{\mathrm{z}}:={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}(\mathsf{\eta }-1).$$

(49)

Note that we [24,25,29] (see also [59]) changed the symbol R for J to avoid the confusion with the Romanovski/pseudo-Jacobi [6,7] polynomials (R-Routh polynomials in our terms [24,25]) denoted in the recent publications [21,60,61,62,63] by the same letter ‘R’. A certain disarray may come from the fact that Koepf and Masjed-Jamei [64,65] used the symbol J for the R-Routh polynomials which is inconsistent with the polynomial names in our classification scheme of the Romanovsi polynomials [10]. Note also that the symbol $\Re$, used for the R-Jacobi polynomials in the recently published paper [66], has been reserved by us [27] for the Routh polynomials [67]:

$${\Re }_{\mathrm{m}}^{({\mathsf{\alpha }}_{\mathrm{R}}+i {\mathsf{\alpha }}_{\mathrm{I}})}(\mathrm{x}):={(-i)}^{\mathrm{m}} {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\alpha }}_{\mathrm{R}}+i {\mathsf{\alpha }}_{\mathrm{I}}, {\mathsf{\alpha }}_{\mathrm{R}}-i {\mathsf{\alpha }}_{\mathrm{I}})}(i\mathrm{x}).$$

(50)

While R-Routh polynomials form finite orthogonal subsequences of the Routh DPS [24], the R-Jacobi polynomials were discovered by Romanovski [30] and have no relation to Routh’s work, contrary to the statement made in [68].

Comparing (48) with (2.8) in [69] shows that the Masjed-Jamei’s M-polynomials are related to the R-Jacobi polynomials (50) via the elementary formula

$${\mathrm{M}}_{\mathrm{n}}^{(\mathrm{p},\mathrm{q})}(\underrightarrow{\mathrm{z}}) ={(-1)}^{\mathrm{n}}\mathrm{n}! {\mathrm{J}}_{\mathrm{n}}^{(\mathrm{q},-\mathrm{p}-\mathrm{q})}(\underrightarrow{\mathrm{z}}) \mathrm{for}\mathrm{q}>\mathrm{-1}, \mathrm{p}>0.$$

(51)

(It is worth to remind the abbreviation ‘OPS’ in [69] stands for the orthogonal polynomial set, but not for the infinite ‘orthogonal polynomial system’—the abbreviation broadly used in the modern theory of the exceptional orthogonal polynomials [15,18,19]).

Substituting q = α, p= −α − β into (2.14) in [69] gives

$${\displaystyle \underset{0}{\overset{\infty }{\int }}\mathrm{d}\underrightarrow{\mathrm{z}}} {\underrightarrow{\mathsf{\omega }}}_{\mathsf{\alpha },\mathsf{\beta }}[\underrightarrow{\mathrm{z}}] {\left|{\mathrm{M}}_{\mathrm{n}}^{(-\mathsf{\alpha }-\mathsf{\beta }, \mathsf{\alpha })} (\underrightarrow{\mathrm{z}})\right|}^2=-\frac{\mathrm{n}!\mathsf{\Gamma }(-\mathsf{\alpha }-\mathsf{\beta }-\mathrm{n})\mathsf{\Gamma }(\mathsf{\alpha }+\mathrm{n}+1)}{(\mathsf{\alpha }+\mathsf{\beta }+2\mathrm{n}+1)\mathsf{\Gamma }(-\mathsf{\beta }-\mathrm{n})}.$$

(52)

As anticipated from (52), the corresponding formula for the R-Jacobi polynomials (see (5) in [70], with $\mathsf{\theta },\mathsf{\vartheta }$ standing for α, β here) differs by the factor (n!)². Indeed, keeping in mind that

$${(-1)}^{\mathrm{n}}{(\mathsf{\beta }+1)}_{\mathrm{n}}\mathsf{\Gamma }(-\mathrm{n}-\mathsf{\beta })=\mathsf{\Gamma }(-\mathsf{\beta }),$$

(53)

one can directly verify that the integral (56), differs exactly by the factor (n!)² from Askey’s [57] original expression for the square integrals of the R-Jacobi polynomials.

It is worth mentioning that our derivation does not cover the orthogonality region of the R-Jacobi polynomials for α varying within the nonpositive interval (−1,0]. This deficiency of our approach dealing solely with eigenfunctions of the p-SLEs may result in some limitations on the actual range of the indexes of the $\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}\mathrm{s}$ of the R-Jacobi polynomials discussed below.

2.5. Quasi-Rational PFSs near the Poles at +1 and Infinity

Let us specify the q-RSs (23) in the slightly way

$${\underset{˜}{\cancel{\mathsf{\psi }}}}_{\stackrel{\rightharpoonup }{\mathsf{\sigma }} ,\mathrm{m}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]:={\underset{˜}{\cancel{\mathsf{\psi }}}}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$$

(54)

with

$${\mathsf{\sigma }}_{\pm }:=sgn{\mathsf{\lambda }}_{\pm }$$

(55)

Our next step is to determine all the q-RSs vanishing at one of the endpoints of the infinite interval (+1, +∞) and then select the subsets of the collected PFSs below the lowest eigenvalue. To explicitly reveal the behavior of the Jacobi-seed (JS) q-RSs (54) near the singular endpoints in question, we [1,24,25], label them as indicated in

Table 1. Classification of JS solutions on the infinite interval (1, ∞) based on their asymptotic behavior near the endpoints.

${\underset{˜}{\mathbf{†}}}_{\stackrel{\rightharpoonup }{\mathsf{\sigma }},m}$	${\mathsf{\sigma }}_{-} {\mathsf{\sigma }}_{+} {\mathsf{\sigma }}_{\infty }$	m
$\underset{˜}{\mathit{a}}$	+   +   −	$0\le \mathrm{m}<\infty$
${\underset{˜}{\mathit{a}}}^{\prime }$	−   +   −	$\mathrm{m}\ge {\mathrm{n}}_{\underset{˜}{\mathit{c}}}={\mathrm{j}}_{\max }+1$
$\underset{˜}{\mathit{b}}$	−   −   +	$0\le \mathrm{m}<{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};-}+{\mathsf{\lambda }}_{\mathrm{o};+}-1)$
${\underset{˜}{\mathit{b}}}^{\prime }$	−   +   +	$0\le \mathrm{m}<{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};+}-{\mathsf{\lambda }}_{\mathrm{o};-}-1)$
$\underset{˜}{\mathit{c}}$	−   +   +	$0\le \mathrm{m}\le {\mathrm{j}}_{\max }=⌊{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};-}-{\mathsf{\lambda }}_{\mathrm{o};+}-1)⌋$
$\underset{˜}{\mathit{d}}$	+   −   −	${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};+}-{\mathsf{\lambda }}_{\mathrm{o};-}-1)\le \mathrm{m}<\infty$
${\underset{˜}{\mathit{d}}}^{\prime }$	−   −   −	$\mathrm{m}>{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};-}+{\mathsf{\lambda }}_{\mathrm{o};+}-1)$

below, with ${\mathsf{\sigma }}_{\pm }$ and ${\mathsf{\sigma }}_{\infty }$ specifying either the decay (+) or growth (−) of the given JS at infinity, i.e., by definition,

$${\underset{˜}{\mathsf{\epsilon }}}_{{\underset{˜}{\mathbf{†}}}_{\stackrel{\rightharpoonup }{\mathsf{\sigma }},\mathrm{m}}, \mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})\equiv -{\mathsf{\epsilon }}_{\stackrel{\rightharpoonup }{\mathsf{\sigma }} ,\mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}}),$$

(56)

where

$${\mathsf{\epsilon }}_{\stackrel{\rightharpoonup }{\mathsf{\sigma }} ,\mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}}):={({\mathsf{\sigma }}_{-}{\mathsf{\lambda }}_{\mathrm{o};-}+{\mathsf{\sigma }}_{+}{\mathsf{\lambda }}_{\mathrm{o};+}+2\mathrm{m}+1)}^2.$$

(57)

In following our olden study [43] on the Darboux transforms ($\mathrm{D}\mathfrak{T}\mathrm{s}$) of radial potentials, we use the letters $\mathit{a}$ and $\mathit{b}$ to specify the PFS near the singular endpoints 1 and ∞ (cases I and II in Quesne’s [71] commonly used classification scheme of q-RSs according to their behavior near the endpoints). We use the letters $\mathit{c}$ and $\mathit{d}$ [43] to identify the ${\mathrm{n}}_{\underset{˜}{\mathit{c}}}$ eigenfunctions and respectively all the q-RSs (54) not vanishing either at +1 or infinity (case III in Quesne’s classification scheme). We underline the corresponding letter by tilde to indicate that the classification of the JS solutions is done on the infinite interval (1, ∞). We mark the letter by the prime symbol if the polynomial components of the given sequence of the q-RSs do not include a constant. (Note that the ‘secondary’ sequences of such a type do not exist for the potentials with infinitely many discrete energy levels which were the focal point of Quesne’s analysis [71]).

Note that the PFSs of the series ${\underset{˜}{\mathit{b}}}^{\prime }$ may exist only if the SLE does not have the discrete energy spectrum. We thus need to consider the three sequences of the quasi-rational PFSs: two primary (starting from m = 0) sequences $\underset{˜}{\mathit{a}}$ and $\underset{˜}{\mathit{b}}$ as well as the infinite secondary sequence ${\underset{˜}{\mathit{a}}}^{\prime }$ starting from $\mathrm{m}={\mathrm{n}}_{\underset{˜}{\mathit{c}}}$.

The primary sequence $\underset{˜}{\mathit{a}}$ is formed by classical Jacobi polynomials and consequently may not have zeros between 1 and ∞. As expected, all the PFSs of this type lie at the energies

$${\underset{˜}{\mathsf{\epsilon }}}_{\underset{˜}{\mathit{a}}, \mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})\equiv -{\mathsf{\epsilon }}_{+ + ,\mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})$$

(58)

below the lowest eigenvalue

$${\underset{˜}{\mathsf{\epsilon }}}_{\underset{˜}{\mathit{c}}, 0}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})\equiv -{\mathsf{\epsilon }}_{+ - ,0}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}}).$$

(59)

The PFSs from the primary sequence $\underset{˜}{\mathit{b}}$ at the energies

$${\underset{˜}{\mathsf{\epsilon }}}_{\underset{˜}{\mathit{b}}, \mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})\equiv -{\mathsf{\epsilon }}_{- - ,\mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})$$

(60)

for

$$0\le \mathrm{m}<{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};-}+{\mathsf{\lambda }}_{\mathrm{o};+}-1)$$

(61)

do not have real zeros larger than 1 iff

$${\underset{˜}{\mathsf{\epsilon }}}_{\underset{˜}{\mathit{b}}, \mathbf{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})-{\underset{˜}{\mathsf{\epsilon }}}_{\underset{˜}{\mathit{c}}, 0}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})=-4({\mathsf{\lambda }}_{\mathrm{o};-}-\mathrm{m}-1)({\mathsf{\lambda }}_{\mathrm{o};+}-\mathrm{m})<0,$$

(62)

i.e., iff

$$0\le \mathbf{m}<{\mathsf{\lambda }}_{\mathrm{o};+}<{\mathsf{\lambda }}_{\mathrm{o};-}-1.$$

(63)

Similarly the PFSs from the secondary sequence ${\underset{˜}{\mathit{a}}}^{\prime }$ at the energies

$${\underset{˜}{\mathsf{\epsilon }}}_{\underset{˜}{{\mathit{a}}^{\prime }}, \mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})= -{\mathsf{\epsilon }}_{+ -, \mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}}) \mathrm{for} \mathrm{m}\ge {\mathrm{n}}_{\underset{˜}{\mathit{c}}}$$

(64)

do not have real zeros larger than 1 iff

$${\underset{˜}{\mathsf{\epsilon }}}_{\underset{˜}{{\mathit{a}}^{\prime }}, \mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})-{\underset{˜}{\mathsf{\epsilon }}}_{\underset{˜}{\mathit{c}}, 0}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathsf{o}})=-4\mathrm{m}({\mathsf{\lambda }}_{\mathrm{o};+}-{\mathsf{\lambda }}_{\mathrm{o};-}+\mathrm{m}+1)<0$$

(65)

or, in other words, iff

$$\mathrm{m}>{\mathsf{\lambda }}_{\mathrm{o};-}-{\mathsf{\lambda }}_{\mathrm{o};+}-1.$$

(66)

Before concluding this section, let us mention that the change of variable (15) turns the quasi-rational functions (21) and (23) into the eigenfunctions of the Schrödinger equation with the h-PT potential, as prescribed by (33) and (34) in [9] with $\mathsf{\alpha }={\mathsf{\lambda }}_{\mathrm{o};+}= {\mathsf{\lambda }}_{+},$ and $\mathsf{\beta }={\mathsf{\lambda }}_{\mathrm{o};-}=-{\mathsf{\lambda }}_{-}.$ Note that the second parameter represents the absolute value of the Jacobi index β defined via (51) above. Also, the h-PT potential (31) in [9] depends on the squares of both Jacobi indexes and therefore one only need to consider the positive values of α, keeping in mind that the potential is repulsive only if α > 1/2. Moreover, all the solutions of the Schrödinger equation with the h-PT potential (31) in [9] are squarely normalizable if 1/2 < α < 1, i.e., the potential has the discrete energy spectrum only for α > 1, i.e., iff the ExpDiff ${\mathsf{\lambda }}_{\mathrm{o};+}\equiv \mathsf{\alpha }$ lies within limit point (LP) range.

3. Use of RZTs for Constructing X_m-Jacobi DPSs

The very important new element of our formalism is the use of RZTs for constructing new RCSLEs which have four infinite sequences of q-RSs. As demonstrated in Section 4.2 below, the polynomial components of these q-RSs obey the Bochner-type ODEs and therefore form four X-Jacobi DPSs.

Let us consider the RRZT with the TF

$${\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]={\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]{\mathsf{\Pi }}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}(\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathrm{m})],$$

(67)

where ${\mathsf{\Pi }}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]$ stands for the monomial product

$${\mathsf{\Pi }}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]:={\displaystyle \prod _{\mathcal{l}=1}^{\mathrm{m}}(\mathsf{\eta }-{\mathsf{\eta }}_{\mathcal{l}})},$$

(68)

with m simple zeros ${\mathsf{\eta }}_{ \mathcal{l}}$, or, in other words, iff the TF has the following rational logarithmic derivative:

$$ld{\mathsf{\varphi }}_m[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}] = ld {\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]+{\displaystyle \sum _{l=1}^{\mathrm{m}}\frac{1}{\mathsf{\eta }-{\mathsf{\eta }}_l(\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathrm{m})}}$$

(69)

By definition of the RZT (see Appendix A for details), the quasi-rational function

$${}^{\ast }\mathsf{\varphi }_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]={\displaystyle \frac{1}{\sqrt{\mathit{\rho } [\mathsf{\eta }] } {\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}}$$

(70)

is the solution of the transformed CSLE

$$\left\{{\displaystyle \frac{{\mathrm{d}}^2 }{\mathrm{d}{\mathsf{\eta }}^2}}+{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]+{\displaystyle \frac{{\mathsf{\epsilon }}_{\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})}{1-{\mathsf{\eta }}^2}} \right\} {}^{\ast }\mathsf{\varphi }_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]= 0$$

(71)

at the energy (25), with the density function defined via (7) for the interval (−1, +1), or at the energy ${\underset{˜}{\mathsf{\epsilon }}}_{\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})$ if the density function is defined via (2) for the interval (1, ∞). Each of the nodeless PFSs below the lowest eigenvalue can be used as the TF to generate the $reg\text{-}\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}$ of the CSLE (1).

Let us now take advantage of the fact that JRef CSL (1) is TFI [72], namely, that quasi-rational function

$${}^{\ast }\mathsf{\varphi }_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]:={\mathit{\rho }}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\xi }] /{\mathsf{\varphi }}_0[\mathsf{\xi };\stackrel{\rightharpoonup }{\mathsf{\lambda }}].$$

(72)

is the solution of the JRef CSLE (1) with ${\mathsf{\lambda }}_{\mathrm{o};\mp }$ replaced for |$|{\mathsf{\lambda }}_{\mp }+1|$, namely,

$${}^{\ast }\mathsf{\varphi }_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]={\mathsf{\varphi }}_0[\mathsf{\eta };-\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}].$$

(73)

Let us also point to another remarkable feature of the JS q-RSs (23)—the Jacobi polynomials in the given sequence are multiplied by the same quasi-rational function and moreover the Jacobi indexes are independent of polynomial degrees [1]. This is the direct consequence of the fact that this CSLE belongs to group A, which is also true for the corresponding Liouville potentials [22].

Let us re-write both RCSLE (71) and

$$\left\{{\displaystyle \frac{{\mathrm{d}}^2 }{\mathrm{d}{\mathsf{\eta }}^2}}+{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]+{\displaystyle \frac{{\mathsf{\epsilon }}_0(\stackrel{\rightharpoonup }{\mathsf{\lambda }})}{1-{\mathsf{\eta }}^2}} \right\} {\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]= 0$$

(74)

in the Riccati form:

$${\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]=-l\stackrel{•}{d} {}^{\ast }\mathsf{\varphi }_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}] -l{\mathrm{d}}^2{}^{\ast }\mathsf{\varphi }_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}] +{\displaystyle \frac{{\mathsf{\epsilon }}_{\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}) }{1-{\mathsf{\eta }}^2}},$$

(75)

and

$${\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]=-l\stackrel{•}{d} {\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}] +l{d}^2{\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}] +{\displaystyle \frac{{\mathsf{\epsilon }}_0(\stackrel{\rightharpoonup }{\mathsf{\lambda }})}{1-{\mathsf{\eta }}^2}},$$

(76)

respectively, where the symbolic expression ld f[η] denotes the logarithmic derivative of the function f[η]. If the density function is identically equal to 1 then the derived expression turns into the standard supersymmetric representation of the quantum mechanical potential in terms of the superpotential represented by the logarithmic derivative of the TF ${\mathsf{\varphi }}_m[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$. In Section 3 we will use a similar representation for the RefPFs of the $\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}\mathrm{s}$ of the JRef CSLE (1) using the quasi-rational TFs (23).

Substituting the q-RS

$${}^{\ast }\mathsf{\varphi }_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]={\mathsf{\varphi }}_0[\mathsf{\eta };-\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}]/{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })$$

(77)

into (75), coupled with (76), one finds

$$\begin{array}{c}{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]= {\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{1}]+2\stackrel{⌢}{\mathrm{Q}}[\mathsf{\eta };\overline{\mathsf{\eta }}(\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathrm{m})]+\hfill \\ 2ld {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })ld {\mathsf{\varphi }}_0[\mathsf{\eta };-\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}]+\left[{\mathsf{\epsilon }}_{\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})-{\mathsf{\epsilon }}_0(-\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1})\right] /({\mathsf{\eta }}^2-1),\hfill \end{array}$$

(78)

where [1,44]

$$\stackrel{⌢}{\mathrm{Q}}[\mathsf{\eta };\overline{\mathsf{\eta }}]:=-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathsf{\Pi }}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{\mathsf{\eta }}^2}}{\mathsf{\Pi }}_{\mathrm{m}}^{-1}[\mathsf{\eta };\overline{\mathsf{\eta }}]=.$$

(79)

$$={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\displaystyle \frac{{\stackrel{••}{\mathsf{\Pi }}}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]}{{\mathsf{\Pi }}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]}}-{\displaystyle \frac{{\stackrel{• }{\mathsf{\Pi }}}_{\mathrm{m}}^2[\mathsf{\eta };\overline{\mathsf{\eta }}]}{{\mathsf{\Pi }}_{\mathrm{m}}^2[\mathsf{\eta };\overline{\mathsf{\eta }}]}}.$$

(80)

It is worth mentioning that the derived expression (78) is valid on the both intervals (1, ∞) and (−1, +1). It can be also trivially extended to two other CSLEs of group A with the Liouville potentials represented by the isotonic oscillator and by Morse potential (assuming that the Schrödinger equation in the latter case is converted to the Bessel-reference CSLE [26]). Making use of (25) and (76), coupled with

$${\mathsf{\epsilon }}_0(\stackrel{\rightharpoonup }{\mathsf{\lambda }})={\mathsf{\epsilon }}_0(-\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1})$$

and

$$ld {\mathsf{\varphi }}_0[\mathsf{\eta };-\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}]=-{\displaystyle \frac{{\mathrm{P}}_1^{({\mathsf{\lambda }}_{+}-1,{\mathsf{\lambda }}_{-}-1)}(\mathsf{\eta })}{{\mathsf{\eta }}^2-1}},$$

(81)

we can re-write (78) as

$$\begin{array}{c}{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]= {\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{1}]+2\stackrel{⌢}{\mathrm{Q}}[\mathsf{\eta };\overline{\mathsf{\eta }}(\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathrm{m})]-\hfill \\ {\displaystyle \frac{2{\mathrm{P}}_1^{({\mathsf{\lambda }}_{+}-1,{\mathsf{\lambda }}_{-}-1)}(\mathsf{\eta }) l\mathrm{d} {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })-\mathrm{m}({\mathsf{\lambda }}_{+}+{\mathsf{\lambda }}_{-}+\mathrm{m}+1)}{{\mathsf{\eta }}^2-1}}.\hfill \end{array}$$

(82)

On other hand, comparing the PF (80) with the Quesne PF

$$\mathrm{Q}[\mathsf{\eta };\overline{\mathsf{\eta }}]:={\displaystyle \frac{{\stackrel{••}{\mathsf{\Pi }}}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]}{{\mathsf{\Pi }}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]}}-{\displaystyle \frac{{\stackrel{• }{\mathsf{\Pi }}}_{\mathrm{m}}^2[\mathsf{\eta };\overline{\mathsf{\eta }}]}{{\mathsf{\Pi }}_{\mathrm{m}}^2[\mathsf{\eta };\overline{\mathsf{\eta }}]}}$$

(83)

(see, e.g., (4.11) in [21], with ${\mathrm{g}}_{\mathrm{m}}^{(\mathrm{A},\mathrm{B})}$ standing for ${\mathsf{\Pi }}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]$ here), one finds

$$\begin{array}{c}\stackrel{⌢}{\mathrm{Q}}[\mathsf{\eta };\overline{\mathsf{\eta }}]=\mathrm{Q}[\mathsf{\eta };\overline{\mathsf{\eta }}]+{\displaystyle \frac{{\stackrel{••}{\mathsf{\Pi }}}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]}{2{\mathsf{\Pi }}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]}}\hfill \\ =-{\displaystyle \sum _{l=1}^{\mathrm{m}}\frac{1}{{(\mathsf{\eta }-{\mathsf{\eta }}_l)}^2}}+{\displaystyle \frac{{\stackrel{••}{\mathsf{\Pi }}}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]}{2{\mathsf{\Pi }}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]}},\hfill \end{array}$$

(84)

In our earlier works [1,44] we overlooked that the Quesne PF (81) is nothing more than

$$\mathrm{Q}[\mathsf{\eta };\overline{\mathsf{\eta }}]=\stackrel{•}{ld}{\mathsf{\Pi }}_{\mathrm{m}}[\mathsf{\eta };\overline{\mathsf{\eta }}]$$

(85)

and therefore does not have simple poles. Taking into account that the trivial identity

$${\mathrm{P}}_1^{({\mathsf{\lambda }}_{+}-1,{\mathsf{\lambda }}_{-}-1)}(\mathsf{\eta })\equiv {\mathrm{P}}_1^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })-\mathsf{\eta },$$

(86)

coupled with the Jacobi equation (26), we can simplify the RefPF in question as follows:

$${\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]= {\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{1}]-{\displaystyle \sum _{l=1}^{\mathrm{m}}\frac{2}{{(\mathsf{\eta }-{\mathsf{\eta }}_l)}^2}}+{\displaystyle \frac{2\mathsf{\eta }}{{\mathsf{\eta }}^2-1}}ld {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta }).$$

(87)

One can easily verify that the derived expression turns into (27) in [29] if we choose m = 1.

The quartet of the Liouville potentials for the RCSLE (71) defined in the four quadrant of the vector parameter $\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ can be thus represented as follows:

$$\begin{array}{c}{\mathrm{V}}_{\stackrel{\rightharpoonup }{\mathsf{\sigma }},\mathrm{m}}[\mathsf{\eta };{}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}}]:=\mathrm{V}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]\hfill \\ =\mathrm{V}[\mathsf{\eta };{}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}}]+|{\mathsf{\eta }}^2-1|\left\{{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]-{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]\right\}\hfill \end{array}$$

(88)

or, using (82),

$$\begin{array}{c}{\mathrm{V}}_{\stackrel{\rightharpoonup }{\mathsf{\sigma }},\mathrm{m}}[\mathsf{\eta };{}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}}]=\mathrm{V}[\mathsf{\eta };{}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}}]-2|{\mathsf{\eta }}^2-1|\stackrel{⌢}{\mathrm{Q}}[\mathsf{\eta };{\overline{\mathsf{\eta }}}^{(\mathrm{m})}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})]+\hfill \\ sgn(\mathsf{\eta }-1)[2{\mathrm{P}}_1^{({\mathsf{\lambda }}_{+}-1,{\mathsf{\lambda }}_{-}-1)}(\mathsf{\eta })ld {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })-\mathrm{m}({\mathsf{\lambda }}_{+}+{\mathsf{\lambda }}_{-}+\mathrm{m}+1)],\hfill \end{array}$$

(89)

with

$${\mathsf{\lambda }}_{\mp }\equiv {\mathsf{\sigma }}_{\mp } {\mathsf{\lambda }}_{\mathrm{o};\mp }, {}^{\ast }\mathsf{\lambda }_{\mathrm{o};\mp }:=|{\mathsf{\sigma }}_{\mp }{\mathsf{\lambda }}_{\mathrm{o};\mp }+1|.$$

(90)

For future references, we made the above expression to be applicable to the Liouville transformations on both intervals (1, ∞) and (−1, +1), which constitutes the essence of the unified approach put forward in [30] for m = 1.

Taking into account that

$$2\underset{\mathsf{\eta }\to \infty }{lim}{\mathsf{\eta }}^2\stackrel{⌢}{\mathrm{Q}}[\mathsf{\eta };\overline{\mathsf{\eta }}]=\mathrm{m}(\mathrm{m}+1)$$

(91)

and

$$2\underset{\mathsf{\eta }\to \infty }{lim}\left\{{\mathrm{P}}_1^{({\mathsf{\lambda }}_{+}-1,{\mathsf{\lambda }}_{-}-1)}(\mathsf{\eta }) ld {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })\right\}=\mathrm{m}({\mathsf{\lambda }}_{+}+ {\mathsf{\lambda }}_{-}),$$

(92)

One can verify that each of the potentials (87) vanishes at infinity, as expected.

Keeping in mind that

$$\stackrel{⌢}{\mathrm{Q}}[\mathsf{\eta };{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})]=-1/{[\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}^2,$$

(93)

for m = 1, we can re-write the corresponding Liouville potential as

$${\mathrm{V}}_{\stackrel{\rightharpoonup }{\mathsf{\sigma }},1}[\mathsf{\eta };{}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}}]=\mathrm{V}[\mathsf{\eta };{}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}}]+sgn({\mathsf{\eta }}^2-1)\left\{{\displaystyle \frac{2({\mathsf{\eta }}^2-1)}{{[\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}^2}}-{\displaystyle \frac{2\mathsf{\eta }}{\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})}}\right\}.$$

(94)

If we set

$$2*\mathrm{A}+1:={}^{\ast }\mathsf{\lambda }_{\mathrm{o};-}-{}^{\ast }\mathsf{\lambda }_{\mathrm{o};+}, 2*\mathrm{B}:={}^{\ast }\mathsf{\lambda }_{\mathrm{o};-}+{}^{\ast }\mathsf{\lambda }_{\mathrm{o};+},$$

(95)

by analogy with (20), and take into account that

$${}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}}=\mp \stackrel{\rightharpoonup }{\mathsf{\lambda }}\mp \stackrel{\rightharpoonup }{1}={\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}\mp \stackrel{\rightharpoonup }{1}\mathrm{for} \stackrel{\rightharpoonup }{\mathsf{\sigma }}=\mp \mp \mathrm{and} {\mathsf{\lambda }}_{\mathrm{o};-}, {\mathsf{\lambda }}_{\mathrm{o};+}>1,$$

(96)

we find that

$$2*\mathrm{A}+1:=\mp ({\mathsf{\lambda }}_{-}-{\mathsf{\lambda }}_{+}), 2*\mathrm{B}:=\mp ({\mathsf{\lambda }}_{-}+{\mathsf{\lambda }}_{+}+2)$$

(97)

$$\mathrm{for} \stackrel{\rightharpoonup }{\mathsf{\sigma }}=\mp \mp ({\mathsf{\lambda }}_{\mathrm{o};-}, {\mathsf{\lambda }}_{\mathrm{o};+}>1 \mathrm{if} \stackrel{\rightharpoonup }{\mathsf{\sigma }}=- -)$$

and therefore

$${\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})= {\displaystyle \frac{2*\mathrm{A}+1}{ 2*\mathrm{B}}}\hspace{1em}\hspace{1em}\mathrm{for} \stackrel{\rightharpoonup }{\mathsf{\sigma }}=\mp \mp ({\mathsf{\lambda }}_{\mathrm{o};-}, {\mathsf{\lambda }}_{\mathrm{o};+}>1 \mathrm{if} \stackrel{\rightharpoonup }{\mathsf{\sigma }}=- -).$$

(98)

Substituting (98) into (94) for $\stackrel{\rightharpoonup }{\mathsf{\sigma }}=\mp \mp$ then gives

$${\mathrm{V}}_{\mp \mp ,1}[\mathsf{\eta };{}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}}]={\mathrm{V}}_{*\mathrm{A},*\mathrm{B}}[\mathsf{\eta }|\mp \mp ,1]:={\mathrm{V}}_{*\mathrm{A},*\mathrm{B}}[\mathsf{\eta }]+$$

(99)

$$sgn({\mathsf{\eta }}^2-1)\left\{{\displaystyle \frac{2(2*\mathrm{A}+1)}{2*\mathrm{B} \mathsf{\eta }-2*\mathrm{A}-1}}-{\displaystyle \frac{4\ast {\mathrm{B}}^2-{(2*\mathrm{A}+1)}^2}{{(2*\mathrm{B} \mathsf{\eta }-2*\mathrm{A}-1)}^2}}\right\}.$$

The change of variable (15) on the interval (1, ∞) then turns (99), with $\stackrel{\rightharpoonup }{\mathsf{\sigma }}=--$, into the potential function (9) in [20], while the change of variable η = sin x converts (99), with $\stackrel{\rightharpoonup }{\mathsf{\sigma }}=++$, into (3.5) in [71], with A and B replaced in both cases for *A and *B respectively.

Let us draw reader’s attention to the fact that the two generally distinct branches of the potential (88) with $\stackrel{\rightharpoonup }{\mathsf{\sigma }}=\mp \mp$ collapse into the same Bagchi-Quesne-Roychoudhury (BQR) potential for m = 1, as expected from the rigorous analysis of the latter case in [29]. The similar collapse takes place for the two other branches of this potential ($\stackrel{\rightharpoonup }{\mathsf{\sigma }}=\mp \pm$) but the resultant potential does not have discrete energy spectrum [29] and therefore cannot be linked to any EOP sequence.

Coming back to the general case m ≥ 1, let us re-write the potential (88) for $\stackrel{\rightharpoonup }{\mathsf{\sigma }}=\mp \mp$ as

$${\mathrm{V}}_{\mp \mp ,m}[\mathsf{\eta };{}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}}]={\mathrm{V}}_{*\mathrm{A},*\mathrm{B}}[\mathsf{\eta }|\mp \mp ,m]:=$$

$$\begin{gathered} {\mathrm{V}}_{*\mathrm{A},*\mathrm{B}}[\mathsf{\eta }]- {\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} (\mp 2*\mathrm{B}+\mathrm{m}+1)|{\mathsf{\eta }}^2-1|\times \\ \left\{(\mp 2*\mathrm{B}+\mathrm{m}){\displaystyle \frac{{\mathrm{P}}_{\mathrm{m}-2}^{({\mathsf{\lambda }}_{+}+2, {\mathsf{\lambda }}_{-}+2)}(\mathsf{\eta })}{{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })}}-(\mp 2*\mathrm{B}+\mathrm{m}+2) {\left[{\displaystyle \frac{{\mathrm{P}}_{\mathrm{m}-1}^{({\mathsf{\lambda }}_{+}+1, {\mathsf{\lambda }}_{-}+1)}(\mathsf{\eta })}{{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })}}\right]}^2\right\}+ \\ (\mp 2*\mathrm{B}+\mathrm{m}+1)sgn(\mathsf{\eta }-1)\left[\mp [(*\mathrm{B}\pm 1)\mathsf{\eta }-*\mathrm{A}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}]{\displaystyle \frac{{\mathrm{P}}_{\mathrm{m}-1}^{({\mathsf{\lambda }}_{+}+1, {\mathsf{\lambda }}_{-}+1)}(\mathsf{\eta })}{{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })}}-\mathrm{m}\right]. \end{gathered}$$

(100)

While our expression (100) for the BQR potential (m = 1) fully agrees with (15) in [11], we detected some discrepancies between (100), with $\stackrel{\rightharpoonup }{\mathsf{\sigma }}=--$, and the corresponding expression for this potential in [11,12].

First, both the potential (19) in [11] and the following expression (37) for the rationally-extended t-PT (Scarf I) potential (after being converted by the change of variable η = sin x (|η| < 1) to its rational form (100) above) lack the term associated with the second derivative of the Jacobi polynomial in the right-hand side of (100). Since this derivative vanishes for the first-degree polynomial, one cannot detect the missed term simply by setting m = 1 in the general formula. The mentioned term was also missed in (81) in [12] or in the preceding expression (64) for the rationally-extended t-PT (Scarf I) potential.

Disregarding the missed term, the rest of both cited expressions in [12] match (100) for $\stackrel{\rightharpoonup }{\mathsf{\sigma }}=--$, if we set

$${\mathsf{\lambda }}_{+}=-\mathsf{\alpha }-1, {\mathsf{\lambda }}_{-}=\mathsf{\beta }-1.$$

(101)

or, taking into account (97),

$$\mathsf{\alpha }=*\mathrm{B}-\mathrm{A}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}, \mathsf{\beta }=-\mathrm{A}-*\mathrm{B}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},$$

(102)

in agreement with the definition of these indexes in [12]. It has been proven above that the potential (88) vanishes at infinity and therefore the function (81) in [12] does not, keeping in mind that the omitted term tends to −m(m − 1) as η → ∞. (The reason for introducing the second pair of the potentials (82) and (83) following (81) in [12] is unclear to me).

As already pointed to in [29], the parameter swap $2\mathrm{B}\leftrightarrow 2\mathrm{A}+1$ does not result in the new potential. The potentials (19) in [11] and (81) in [12] differ only by notation, with the constraint $2\mathrm{B}>2\mathrm{A}+1$ reversed for $2\mathrm{B}<2\mathrm{A}+1$. This also true for he two potentials (12) and (15) listed in [11] for m = 1.

4. Pseudo-Wronskian Representation of X_m-Jacobi DPSs

Based on Rudyak and Zakhariev’s [45] generic formula for the solutions of the transformed CSLE the general solution of the RCSLE

$$\left\{{\displaystyle \frac{{\mathrm{d}}^2 }{\mathrm{d}{\mathsf{\eta }}^2}}+{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]+\underset{˜}{\mathsf{\epsilon }} \rho [\mathsf{\eta }] \right\} \mathsf{\Phi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};\underset{˜}{\mathsf{\epsilon }}|\mathrm{m}]= 0$$

(103)

can be represented as

$$\mathsf{\Phi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};\underset{˜}{\mathsf{\epsilon }}|\mathrm{m}]\equiv \mathsf{\Phi }[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\underset{˜}{\mathsf{\epsilon }}|\stackrel{\rightharpoonup }{\mathsf{\sigma }}, \mathrm{m}]:={\displaystyle \frac{W\{{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ],\mathsf{\Phi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ;\underset{˜}{\mathsf{\epsilon }}]\}}{{\rho }^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} [\mathsf{\eta }]{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}}$$

(104)

$$={\rho }^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} [\mathsf{\eta }]\stackrel{•}{\mathsf{\Phi }}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ;\underset{˜}{\mathsf{\epsilon }}]-{\rho }^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} [\mathsf{\eta }]ld{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]\mathsf{\Phi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ;\underset{˜}{\mathsf{\epsilon }}].$$

(105)

The gauge transformation

$$\mathsf{\Psi } [\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}; \underset{˜}{\mathsf{\epsilon }}|\mathrm{m}]= {\rho }^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}} [\mathsf{\eta }] \mathsf{\Phi } [\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}; \underset{˜}{\mathsf{\epsilon }}|\mathrm{m}]$$

(106)

converts the RCSLE (103) into the algebraic Schrödinger equation

$$\left\{{\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}\sqrt{{\mathsf{\eta }}^2-1}{\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}+\left(\underset{˜}{\mathsf{\epsilon }}-\mathrm{V}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]\right)\sqrt{{\mathsf{\eta }}^2-1}\right\} \mathsf{\Psi } [\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}; \underset{˜}{\mathsf{\epsilon }}|\mathrm{m}]= 0.$$

(107)

Substituting (106) into (104) then gives

$$\mathsf{\Psi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};\underset{˜}{\mathsf{\epsilon }}|\mathrm{m}]\equiv \mathsf{\Psi }[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\underset{˜}{\mathsf{\epsilon }}|\stackrel{\rightharpoonup }{\mathsf{\sigma }}, \mathrm{m}]:={\displaystyle \frac{W\{{\mathsf{\psi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ],\mathsf{\Psi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ;\underset{˜}{\mathsf{\epsilon }}]\}}{{\rho }^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} [\mathsf{\eta }]{\mathsf{\psi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}}$$

(108)

$$={\rho }^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} [\mathsf{\eta }]\stackrel{•}{\mathsf{\Psi }}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ;\underset{˜}{\mathsf{\epsilon }}]+w_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]\mathsf{\Psi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ;\underset{˜}{\mathsf{\epsilon }}]$$

(109)

where

$$w_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]:=-{\rho }^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]ld{\mathsf{\psi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}].$$

(110)

Note that the latter function is holomorphic if the square-root of the density function in the JRef CSLE (1) is a PF. As a result, the RS function becomes the holomorphic ‘prepotential’ in Ho’s terms [73,74,75,76,77]. In particular [75], the Rosen-Morse [78] potential represents the simplest case when ${\mathit{\rho }}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]$ is the first-degree polynomial [55]. The latter requirement also holds if the numerator of the PF $\mathit{\rho }[\mathsf{\eta }]$ has a double zero [79], with the Manning-Rosen [80] potential (‘Eckart’ potential in [81] as its limiting case).

The representation of the general solution of the algebraic Schrödinger equation in the form (109) provides the accurate mathematical basis for the conventional SUSY theory of rationally extended potentials utilized in [12,20,30,71]. It should be noticed in this connection that the renowned SUSY rules [53] for the changes in energy spectra due to DTs of the Schrödinger equation were originally formulated [38,39,81,82] for potentials with exponential tails at ±∞. The latter restriction forced the ODE under consideration to have the LP singularities at both quantization ends. The formulated rules were then extended by Sukumar [83] to radial potentials without properly treating the LC range of the centrifugal barrier. Since then, the SUSY rules for the radial Schrödinger equation were duplicated without the proper examination of this non-trivial problem, with Gangopadhyaya et al.’s paper [84] and Chapter 12 in [85] as the only known-to-us exceptions.

If the ExpDiff for the pole at the origin lies within the LC range any solution is square integrable and the reg-RRZT with the TF of type $\stackrel{\rightharpoonup }{\mathsf{\sigma }}=--$ does not results in the isospectral problem if the ExpDiff for the pole of the JRef CSLE (1) lies between 1 and 2. On the other hand, as discussed in more detail in Section 5.3, we can still construct the corresponding EOP sequence using the polynomial components of the quasi-rational eigenfunctions of the p-SLE solved under the DBCs.

4.1. $RRZ\mathfrak{T}s$ of JS Solutions

The main purpose of this subsection is to demonstrate that the RCSLE (103) has four infinite sequences of the q-RSs with the polynomial components represented by either the Wronskians $({\cancel{\mathsf{\sigma }}}_{\mp }=+)$ or $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}\mathrm{s}$$(\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}\ne + +)$ of two Jacobi polynomials. It will be then proven in the following subsection, that these polynomials obey the Bochner-type ODEs and therefore form a X-Jacobi DPS. As mentioned in Introduction, this is one of the core elements of our technique making it possible to cover both infinite and finite EOP sequences.

To achieve the sketched result, let us consider in parallel the four $\mathrm{R}\mathrm{D}\mathrm{C}\mathfrak{T}\mathrm{s}$ of the q-RS

$${\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]\equiv {\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}^{\prime }\times {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}],$$

(111)

using the TFs

$${\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]\equiv {\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\sigma }}\times {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}],$$

(112)

where $\stackrel{\rightharpoonup }{\mathsf{\sigma }}$ and ${\stackrel{\rightharpoonup }{\mathsf{\sigma }}}^{\prime }$ specify the quadrants of the vectors $\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ and ${\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }$ respectively. By definition,

$${\mathsf{\lambda }}_{\mp }^{\prime }:={\cancel{\mathsf{\sigma }}}_{\mp }{\mathsf{\lambda }}_{\mp } ,$$

(113)

where ${\cancel{\mathsf{\sigma }}}_{\mp }$ is equal to either + or-and

$$|{\mathsf{\lambda }}_{\pm }^{\prime }|=|{\mathsf{\lambda }}_{\pm }|={\mathsf{\lambda }}_{\mathrm{o};\pm }.$$

(114)

We are now in position to prove that the corresponding solutions (104) of the RCSLE (103) at the common energy

$$\underset{˜}{\mathsf{\epsilon }}=-{\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })$$

have the quasi-rational form with the polynomial components mentioned at the beginning of the section.

Theorem 1.

RCSLE (103) has four infinite sequences of q-RSs, with the polynomial components represented by polynomial Wronskians for one of these sequences and by p-Ws of two Jacobi polynomials for three others.

Proof. 

In the simplest case $\stackrel{\rightharpoonup }{\mathsf{\lambda }} ={\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }$ ($\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=+ +$) the Wronskian of the two JS solutions takes form

$$\mathrm{W}\{{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}],{\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]\}={\mathsf{\varphi }}_0^2[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]{\mathrm{W}}_{\mathrm{m}+\mathrm{j}-1}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}[\mathsf{\eta }|\mathrm{m},\mathrm{j}],$$

(115)

where the polynomial of degree m + j − 1 in the right-hand side stands for the Wronskian of two Jacobi polynomials with the same pair of the indexes:

$${\mathrm{W}}_{\mathrm{m}+\mathrm{j}-1}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}[\mathsf{\eta }|\mathrm{m},\mathrm{j}]:=\mathrm{W}\{{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta }),{\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })\},$$

(116)

Next, taking into account that [51]

$$\begin{gathered} |1-\aleph \mathsf{\eta }{|}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\cancel{\mathsf{\sigma }}}_{\aleph }1-1){\mathsf{\lambda }}_{\aleph }^{\prime }}{\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}[{\displaystyle \prod _{\aleph =\mp }|}1-\aleph \mathsf{\eta }{|}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left({\mathsf{\lambda }}_{\aleph }^{\prime }-{\cancel{\mathsf{\sigma }}}_{\aleph }{\mathsf{\lambda }}_{\aleph }^{\prime }\right)}{\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta })]= \\ {\mathrm{d}}_{\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}},\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}){\displaystyle \prod _{\aleph =\mp }{(\mathsf{\eta }-\aleph 1)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.({\cancel{\mathsf{\sigma }}}_{\aleph }1-1)}}{\mathrm{P}}_{\mathrm{j}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}^{({\mathsf{\lambda }}_{+}^{\prime }+{\cancel{\mathsf{\sigma }}}_{+}1, {\mathsf{\lambda }}_{-}^{\prime }+{\cancel{\mathsf{\sigma }}}_{-}1)}(\mathsf{\eta }), \end{gathered}$$

(117)

where

$${\mathrm{d}}_{\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}},\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })=\left\{\begin{array}{ll} \mathrm{j}+{\mathsf{\lambda }}_{+}^{\prime }& \mathrm{i}\mathrm{f} \cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=+ -,\\ \mathrm{j}+ {\mathsf{\lambda }}_{-}^{\prime }& \mathrm{i}\mathrm{f} \cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}= - +, \\ 2(\mathrm{j}+1)& \mathrm{i}\mathrm{f} \cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}= - -, \\ {\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}(\mathrm{j}+{\mathsf{\lambda }}_{+}^{\prime }+{\mathsf{\lambda }}_{-}^{\prime }+1)& \mathrm{i}\mathrm{f} \cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=+ +, \end{array}\right.$$

(118)

(see (91) in [51] with $\mathsf{\alpha }={\mathsf{\lambda }}_{+}^{\prime }, \mathsf{\beta }={\mathsf{\lambda }}_{-}^{\prime }$, and n replaced for j), we introduce the $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}\mathrm{s}$ via the relations:

$$\begin{gathered} {\mathfrak{P}}_{\mathrm{m}+\mathrm{j}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{m}; \mathrm{j}]:= \\ \left|\begin{array}{cc}{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })& {\displaystyle \prod _{\aleph =\mp }(}\mathsf{\eta }-\aleph 1{)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(1-{\cancel{\mathsf{\sigma }}}_{\aleph }\right)} {\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta })\\ {\stackrel{•}{\mathrm{P}}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })& {\mathrm{d}}_{\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}},\mathrm{m}}\mathrm{j}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }){\mathrm{P}}_{\mathrm{j}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}^{({\mathsf{\lambda }}_{+}^{\prime }+{\cancel{\mathsf{\sigma }}}_{+}1, {\mathsf{\lambda }}_{-}^{\prime }+{\cancel{\mathsf{\sigma }}}_{-}1)}(\mathsf{\eta })\end{array}\right| \end{gathered}$$

(119)

and then re-write the Wronskian of the q-RSs (110) and (111),

$$\begin{array}{c}\mathrm{W}\{{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}],{\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]\}={\mathsf{\varphi }}_0^2[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}] \times \hfill \\ \mathrm{W}\{{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta }), |1-\aleph \mathsf{\eta }{|}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\aleph }^{\prime }-{\mathsf{\lambda }}_{\aleph })}{\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta })\},\hfill \end{array}$$

(120)

as follows:

$$\begin{gathered} \mathrm{W}\{{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}],{\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]\}= {\mathsf{\varphi }}_0[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{1}-\stackrel{\rightharpoonup }{1}]{\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]\times \\ {\mathfrak{P}}_{\mathrm{m}+\mathrm{j}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{m}; \mathrm{j}]. \end{gathered}$$

(121)

For $\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=+ +$ the latter expression turns into (115) with

$${\mathfrak{P}}_{\mathrm{m}+\mathrm{j}-1}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|+ +,\mathrm{m}; \mathrm{j}]={\mathrm{W}}_{\mathrm{m}+\mathrm{j}-1}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}[\mathsf{\eta }|\mathrm{m},\mathrm{j}].$$

(122)

Making use of (121), the RRZT of the q-RS (111),

$$\mathsf{\Phi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};-{\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })|\mathrm{m}]=\sqrt{|1-{\mathsf{\eta }}^2|}{\displaystyle \frac{\mathrm{W}\{{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ],{\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]\}}{{\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })}},$$

(123)

can be re-written in the following quasi-rational form:

$$\mathsf{\Phi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};-{\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })|\mathrm{m}]= {\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}] {\mathfrak{P}}_{\mathrm{m}+\mathrm{j}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{m};\mathrm{j}]$$

(124)

with

$${\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}]:={\displaystyle \frac{ {\mathsf{\varphi }}_0[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{1}]}{{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })}},$$

(125)

which completes the proof of the Theorem 1. □

Representing the polynomials (119) as

$$\begin{gathered} {\mathfrak{P}}_{\mathrm{m}+\mathrm{j}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{m}; \mathrm{j}]:= \\ \left|\begin{array}{c}{\displaystyle \prod _{\aleph =\mp }(}\mathsf{\eta }-\aleph 1{)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(1-{\cancel{\mathsf{\sigma }}}_{\aleph }\right)} {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta }) {\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta })\\ {\mathrm{d}}_{\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}},\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}){\mathrm{P}}_{\mathrm{m}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}^{({\mathsf{\lambda }}_{+}+{\cancel{\mathsf{\sigma }}}_{+}1, {\mathsf{\lambda }}_{-}+{\cancel{\mathsf{\sigma }}}_{-}1)}(\mathsf{\eta }) {\stackrel{•}{\mathrm{P}}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta })\end{array}\right|. \end{gathered}$$

(126)

and choosing $\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=\mp \pm$ in (119), we come to the two infinite polynomial sequences:

$$\begin{array}{c}{\mathfrak{P}}_{\mathrm{j}+\mathrm{m}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }| \mp \pm ,\mathrm{m}; \mathrm{j}]=(\mathsf{\eta }\pm 1){\mathrm{P}}_{\mathrm{m}}^{(\pm {\mathsf{\lambda }}_{+}^{\prime }, \mp {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta }){\stackrel{•}{\mathrm{P}}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta })+\hfill \\ ({\mathsf{\lambda }}_{\mp }^{\prime }-\mathrm{m}){\mathrm{P}}_{\mathrm{m}}^{(\pm {\mathsf{\lambda }}_{+}^{\prime }\pm 1,\mp {\mathsf{\lambda }}_{-}^{\prime }\mp 1)}(\mathsf{\eta }){\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta })\hfill \end{array}$$

(127)

which constitute the core of this discussion. Though the two are interrelated via the interchange of the two indexes accompanied by the change in the argument:

$${\mathfrak{P}}_{\mathrm{j}+\mathrm{m}}[\mathsf{\eta };{\mathsf{\lambda }}_{-}^{\prime },{\mathsf{\lambda }}_{+}^{\prime }|- +,\mathrm{m};\mathrm{j}]={(-)}^{\mathrm{j}+\mathrm{m}+1}{\mathfrak{P}}_{\mathrm{j}+\mathrm{m}}[-\mathsf{\eta };{\mathsf{\lambda }}_{+}^{\prime },{\mathsf{\lambda }}_{-}^{\prime }|+ -.\mathrm{m};\mathrm{j}],$$

(128)

we (in contrast with [15,16]) prefer to treat these two polynomial sequences independently and refer to them as being of series J1 and J2, in following the terminology suggested in [3]. Note that, contrary to the definition (2.3) of these two series in [3], with m = n,

$$\mathrm{J}1: {\mathsf{\lambda }}_{-}^{\prime }={\mathsf{\lambda }}_{\mathrm{o};-}\equiv h+\mathrm{m}+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}>0, {\mathsf{\lambda }}_{+}^{\prime }={\mathsf{\lambda }}_{\mathrm{o};+}\equiv g+\mathrm{m}-{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}>0,$$

and

$$\mathrm{J}2: {\mathsf{\lambda }}_{-}^{\prime }={\mathsf{\lambda }}_{\mathrm{o};-}\equiv h+\mathrm{m}-{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}>0, {\mathsf{\lambda }}_{+}^{\prime }={\mathsf{\lambda }}_{\mathrm{o};+}\equiv g+\mathrm{m}+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}>0$$

in our notation, the indexes of the Jacobi polynomials in the right-hand side of (128) are independent of the polynomial degrees. It will be proven below that the polynomial sequences (127) satisfy the Bochner-type ODEs but violate the Bochner theorem, since they do not start from a constant. We [24] refer to them as the X_m-Jacobi DPSs of series J1 and J2, with no restrictions imposed on the Jacobi indexes other than their absolute values differ from 0 and 1.

In particular, choosing $\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=+ -$ and setting $\mathsf{\beta }={\mathsf{\lambda }}_{-}^{\prime }, \mathsf{\alpha }={\mathsf{\lambda }}_{+}^{\prime }$ in (119) and (127) we come to (72) and (71) in [16]:

$${\mathfrak{P}}_{\mathrm{m}+\mathrm{j}}[\mathsf{\eta };\mathsf{\beta },\mathsf{\alpha }|+ -,\mathrm{m}; \mathrm{j}]=(\mathsf{\alpha }+\mathrm{j}){\mathrm{P}}_{\mathrm{j}}^{(\mathsf{\alpha }-1, \mathsf{\beta }+1)}(\mathsf{\eta }){\mathrm{P}}_{\mathrm{m}}^{(-\mathsf{\alpha }, \mathsf{\beta })}(\mathsf{\eta })$$

(129)

$$-(\mathsf{\eta }-1){\mathrm{P}}_{\mathrm{j}}^{(\mathsf{\alpha }, \mathsf{\beta })}(\mathsf{\eta }) {\stackrel{•}{\mathrm{P}}}_{\mathrm{m}}^{(-\mathsf{\alpha }, \mathsf{\beta })}(\mathsf{\eta })$$

$$=({\mathsf{\alpha }}_{+}-\mathrm{m}){\mathrm{P}}_{\mathrm{m}}^{(-\mathsf{\alpha }-1,\mathsf{\beta }+1)}(\mathsf{\eta }){\mathrm{P}}_{\mathrm{j}}^{(\mathsf{\alpha }, \mathsf{\beta })}(\mathsf{\eta })$$

$$+(\mathsf{\eta }-1){\mathrm{P}}_{\mathrm{m}}^{(-\mathsf{\alpha }, \mathsf{\beta })}(\mathsf{\eta }){\stackrel{•}{\mathrm{P}}}_{\mathrm{j}}^{(\mathsf{\alpha }, \mathsf{\beta })}(\mathsf{\eta })$$

(130)

and, consequently,

$${\mathfrak{P}}_{\mathrm{j}+\mathrm{m}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|+ -,\mathrm{m};\mathrm{j}]={(-1)}^{\mathrm{m}}({\mathsf{\lambda }}_{+}^{\prime }+\mathrm{j}){\stackrel{⌢}{\mathrm{P}}}_{\mathrm{m},\mathrm{m}+\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime }-1,{\mathsf{\lambda }}_{-}^{\prime }+1)}(\mathsf{\eta }),$$

(131)

with no limitation on the indexes ${\mathsf{\lambda }}_{\mp }^{\prime }$ in the given context.

For m = 1 the two X-DPSs collapse into the single X-DPS of series J [29], which in particular contains the X₁-Jacobi OPS [31,32] often referred to simply as ‘X₁-Jacobi orthogonal polynomials’. The term ‘X₁-Jacobi polynomials’ introduced by Yadav et al. [23] (while referring to the X₁-Jacobi DPS of series J) seems satisfactory as far as one clearly understands that that that we deal here with a larger family of polynomials which contains both the X₁-Jacobi OPS and the only existent finite EOP sequence [29].

On other hand, coming back to (101) and setting:

$${\mathsf{\lambda }}_{-}= {\mathsf{\lambda }}_{-}^{\prime }=\mathsf{\beta }-1=-{\mathsf{\lambda }}_{\mathrm{o};-}, -{\mathsf{\lambda }}_{+}={\mathsf{\lambda }}_{+}^{\prime }=\mathsf{\alpha }+1={\mathsf{\lambda }}_{\mathrm{o};+}$$

(132)

in (127) for $\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=+ -$, one finds

$$\begin{gathered} {\mathfrak{P}}_{\mathrm{j}+\mathrm{m}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|+ -,\mathrm{m}; \mathrm{j}]=({\mathsf{\lambda }}_{+}^{\prime }-\mathrm{m}){\mathrm{P}}_{\mathrm{m}}^{(-{\mathsf{\lambda }}_{+}^{\prime }-1,{\mathsf{\lambda }}_{-}^{\prime }+1)}(\mathsf{\eta }){\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta }) \\ -{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{+}^{\prime }+ {\mathsf{\lambda }}_{-}^{\prime }+\mathrm{j}+1)(1-\mathsf{\eta }){\mathrm{P}}_{\mathrm{m}}^{(-{\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta }){\mathrm{P}}_{\mathrm{j}-1}^{({\mathsf{\lambda }}_{+}^{\prime }+1, {\mathsf{\lambda }}_{-}^{\prime }+1)}(\mathsf{\eta }), \end{gathered}$$

(133)

which brings us to the polynomial sequence (15) in [10] (with ν standing for j here). Note that (132) is consistent the m-independent definition of the indexes α and β in [11]. With the latter correction, (15) in [10] turns into (133) here, which is nothing but the X_m-Jacobi DPS of series J2 in our terms. It does contain both the X_m-Jacobi OPS [15,16] and the finite EOP sequences of type $\tilde{\mathit{b}}$ used by Yadav et al. [10,11,12,13] to obtain the analytical expression for the eigenfunctions of the rationally extended h-PT potential of the same type (and referred to in the cited papers simply as ‘X_m-Jacobi polynomials’).

Similarly, the ladders of lowering and raising operators introduced by Yadav et al. [14,86] act in the space spanned by the X_m-Jacobi DPS of series J2. The title ‘X_m-Jacobi orthogonal polynomials’ used for the subsection IV.A in [14] seems misleading since the ladder relations analyzed by the cited authors are applicable for any values of the indexes α and β.

The polynomials belong to the X-OPS iff the indexes α and β are larger than -1 and have the same sign. The lowering operators necessarily bring these indexes into a region where the polynomials do not belong to the X-OPS anymore. We postpone the thorough analysis of this issue for a separate publication.

Examination of the scattering amplitude (25) in [10] reveals that it has poles on the imaginary positive k-axis at

$$k=(\mathrm{A}-\mathrm{j})i.$$

(134)

By choosing

$${\mathsf{\lambda }}_{\mp }=-{\mathsf{\lambda }}_{\mathrm{o};\mp } (\stackrel{\rightharpoonup }{\mathsf{\sigma }}=- -),{\mathsf{\lambda }}_{\mp }^{\prime }=\mp {\mathsf{\lambda }}_{\mathrm{o};\mp }$$

(135)

and taking into account that, according to (97),

$$2*\mathrm{A}+1=2\mathrm{A}+1=-{\mathsf{\lambda }}_{-}^{\prime }-{\mathsf{\lambda }}_{+}^{\prime }\mathrm{for} \stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=+ -,$$

(136)

we confirm that the mentioned poles correspond to the bound energies (40), as expected. We thus assume that the cited scattering amplitude was obtained in [10] using the constraint 2B > 2A + 1, but not the change of the notation suggested by the cited authors later in writing the potential function (19) in [11].

Let us now consider the alternative quasi-rational representation of the Wronskian (121):

$$\mathrm{W}\{{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}],{\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]\}={\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}]{\mathsf{\varphi }}_0[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }-\stackrel{\rightharpoonup }{1}]{\mathfrak{D}}_{\mathrm{m}+\mathrm{j}+1}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }},\mathrm{m};{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime },\mathrm{j}],$$

(137)

in terms of the so-called [48] ‘polynomial determinant’ (PD)

$${\mathfrak{D}}_{\mathrm{m}+\mathrm{j}+1}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }},\mathrm{m};{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime },\mathrm{j}]:=\left|\begin{array}{c}{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta }) {\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta })\\ {\mathrm{S}}_{\mathrm{m}+1}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta }) {\mathrm{S}}_{\mathrm{j}+1}^{( {\mathsf{\lambda }}_{+}^{\prime }, {\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta }) \end{array}\right|,$$

(138)

where

$${\mathrm{S}}_{\mathrm{m}+1}^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta }):={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}[(\mathsf{\alpha }+1)(\mathsf{\eta }+1)+(\mathsf{\beta }+1)(\mathsf{\eta }-1)] {\mathrm{P}}_{\mathrm{m}}^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta })+({\mathsf{\eta }}^2-1){\stackrel{•}{\mathrm{P}}}_{\mathrm{m}}^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta }).$$

(139)

Originally [8,28] we used the PD decomposition as the universally applicable starting point for representing the $\mathrm{R}\mathrm{D}\mathrm{C}\mathfrak{T}\mathrm{s}$ of JS solutions in the quasi-rational form for any admissible density function (see [87] for an example). Yet the same year Gómez-Ullate et al. [51] suggested a powerful scheme allowing one to represent the $\mathrm{R}\mathrm{D}\mathrm{C}\mathfrak{T}\mathrm{s}$ of JS solutions in the quasi-rational form, with the polynomial components represented by the $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}\mathrm{s}$. The main advantage of using the $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}\mathrm{s}$ (instead of PDs) for the $\mathrm{R}\mathrm{D}\mathfrak{T}\mathrm{s}$ of JS solutions discussed here is that the $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}\mathrm{s}$, in contrast with PDs, remain finite at both singular points $\mp 1$ and as a result form the X_m-Jacobi DPSs. Though we need to mention that this assertion does not generally retain for the polynomial components of the higher-order $\mathrm{R}\mathrm{D}\mathrm{C}\mathfrak{T}\mathrm{s}$ in their $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}$ representation.

Comparing (137) with (121), one finds that

$$\begin{array}{c}{\mathfrak{D}}_{\mathrm{m}+\mathrm{j}+1}[\mathsf{\eta };\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime },\mathrm{m};{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime },\mathrm{j}]={(\mathsf{\eta }-1)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\cancel{\mathsf{\sigma }}}_{-}+1)}{(\mathsf{\eta }-1)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\cancel{\mathsf{\sigma }}}_{+}+1)}\times \hfill \\ {\mathfrak{P}}_{\mathrm{m}+\mathrm{j}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{m}; \mathrm{j}]\hfill \end{array}$$

(140)

It directly follows from (140) that the PD and $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}$ representations become identical for $\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=--$. Evaluating the polynomials (139) at $\mp 1$:

$${\mathrm{S}}_{\mathrm{m}+1}^{(\mathsf{\alpha },\mathsf{\beta })}(\mp 1)={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}[(\mathsf{\alpha }+1)(\mp 1+1)+(\mathsf{\beta }+1)(\mp 1-1)] {\mathrm{P}}_{\mathrm{m}}^{(\mathsf{\alpha },\mathsf{\beta })}(\mp 1)$$

(141)

and substituting (141) into the right-hand side of (138) gives

$${\mathfrak{D}}_{\mathrm{j}+2}[\mp 1;-{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime },1;{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime },\mathrm{j}]=\mp 2{\mathsf{\lambda }}_{\mp }^{\prime }{\mathrm{P}}_1^{(-{\mathsf{\lambda }}_{+}^{\prime },-{\mathsf{\lambda }}_{-}^{\prime })}(\mp 1){\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{+}^{\prime },{\mathsf{\lambda }}_{-}^{\prime })}(\mp 1).$$

(142)

We have thus proven that the PD (138) remains finite at $\mp 1$, provided that this is true for the j-th degree Jacobi polynomial on the right. Out of the four PD sequences introduced by us in [28] to generate the X_m-Jacobi DPSs, this was the only sequence which retains finite at ±1 and therefore spanned the DPS referred to by us for this reason as the X_m-Jacobi DPS of series D.

4.2. Bochner-Type ODEs for Four X_m-Jacobi DPSs

Our next step is to prove that both polynomial Wronskian (116) and all three $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}\mathrm{s}$ forming the q-RS (124) with $\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\ne + +$ satisfy the Bochner-type ODEs and as a result form four X_m-Jacobi DPSs.

Before proceeding with the proof, let us first draw reader’s attention to the fact that the magnitudes of the exponent parameters of the basic solution ${\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times (\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{1})]$ in the right hand-side of (125)

$$|{\mathsf{\lambda }}_{\pm }^{\prime }+{\cancel{\mathsf{\sigma }}}_{\pm }1| ={}^{\ast }\mathsf{\lambda }_{\mathrm{o};\pm }:= |{\mathsf{\lambda }}_{\pm }+1|$$

(143)

represent the ExpDiffs for the poles of the RCSLE (103) at ±1 which implies that that each $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}$ remains finite at both poles at least if ${\mathsf{\lambda }}_{\pm }+1$ are not negative integers with the absolute values smaller than the polynomial degree.

The transition from the RCSLE (103) to the Bochner-type ODE with polynomial coefficients is completed by the gauge transformation with the gauge function (125), which, as one can easily verify, satisfies the RCSLE

$${\stackrel{••}{\mathsf{\varphi }}}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}]+\left\{{\mathrm{I}}_0^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]-{\displaystyle \frac{{\mathsf{\epsilon }}_0({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+ \cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{1})}{{\mathsf{\eta }}^2-1}}\right\}{\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}]=0$$

(144)

with the RefPF

$$\begin{array}{c}{\mathrm{I}}_0^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|m]:={\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{1}]+2\stackrel{⌢}{\mathrm{Q}}[\mathsf{\eta };{\overline{\mathsf{\eta }}}^{(\mathrm{m})}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m})]\hfill \\ +2ld{\mathsf{\varphi }}_0[\mathsf{\eta };\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}\times (\stackrel{\rightharpoonup }{\mathsf{\lambda }}+ \stackrel{\rightharpoonup }{1})]ld {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta }).\hfill \end{array}$$

(145)

The crucial point for our analysis is that the second-order poles in the difference between the RefPFs (82) and (145) compensate each other. As a result, substituting (124) into the RCSLE (103) brings us to the Bochner-type ODE:

$$\{{\mathbf{D}}_{\mathsf{\eta }}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}})+C_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};{\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })|\mathrm{m};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}} ]\} {\mathfrak{P}}_{\mathrm{m}+\mathrm{j}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{m};\mathrm{j}]=0,$$

(146)

where ${\mathbf{D}}_{\mathsf{\eta }}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}})$ is the abbreviated notation for the second-order differential operator in η,

$${\mathbf{D}}_{\mathsf{\eta }}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}|m;\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}):=(1-{\mathsf{\eta }}^2){\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta }){\displaystyle \frac{{\mathrm{d}}^2 }{\mathrm{d}{\mathsf{\eta }}^2}}+2{\mathrm{B}}_{\mathrm{m}+1}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}]{\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}.$$

(147)

Keeping in mind that

$$ld {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })={\displaystyle \sum _{l=1}^{\mathrm{m}} \frac{1}{\mathsf{\eta }-{\mathsf{\eta }}_l(\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathrm{m})}},$$

(148)

we find that the coefficient function of the first derivative is represented by the following polynomial of degree m + 1:

$$\begin{gathered} {\mathrm{B}}_{\mathrm{m}+1}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}]:= (1-{\mathsf{\eta }}^2) {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta }) \\ \times \left({\displaystyle \sum _{\aleph =\pm }\frac{{\cancel{\mathsf{\sigma }}}_{\aleph }({\mathsf{\lambda }}_{\aleph }+1)+1}{2(\mathsf{\eta }-\aleph 1)}}-{\displaystyle \sum _{l=1}^{\mathrm{m}} \frac{1}{\mathsf{\eta }-{\mathsf{\eta }}_l(\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathrm{m})}}\right) \end{gathered}$$

(149)

and also that the ε-dependent polynomial of degree m being regarded as the free term of the ODE (142) is linear in the energy:

$${\mathrm{C}}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathsf{\epsilon }|\mathrm{m};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}} ]={\mathrm{C}}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}} ]-\mathsf{\epsilon } {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta }).$$

(150)

Substituting (82) and (145) into the definition of the zero-energy part of (150):

$${\mathrm{C}}_{ \mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}]=({\mathsf{\eta }}^2-1)\left\{{\mathrm{I}}^{\mathrm{o}} [\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]-{\mathrm{I}}_0^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]\right\}-{\mathsf{\epsilon }}_0({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+ \cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{1}){\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta }),$$

one finds

$$\begin{gathered} {\mathrm{C}}_{ \mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}]=2\left[{\mathrm{P}}_1^{({\mathsf{\lambda }}_{+}+1,{\mathsf{\lambda }}_{-}+1)}(\mathsf{\eta })-{\mathrm{P}}_1^{({\mathsf{\lambda }}_{+}^{\prime }+{\cancel{\mathsf{\sigma }}}_{+}1,{\mathsf{\lambda }}_{-}^{\prime }+{\cancel{\mathsf{\sigma }}}_{-}1)}(\mathsf{\eta })\right] \\ \times {\stackrel{•}{\mathrm{P}}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })+\left[\mathrm{m}({\mathsf{\lambda }}_{+}+{\mathsf{\lambda }}_{-}+\mathrm{m}+1)-{\mathsf{\epsilon }}_0({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+ \cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{1})\right] {\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta }), \end{gathered}$$

(151)

where

$$\begin{array}{c}{\mathrm{P}}_1^{({\mathsf{\lambda }}_{+}+1,{\mathsf{\lambda }}_{-}+1)}(\mathsf{\eta })-{\mathrm{P}}_1^{({\mathsf{\lambda }}_{+}^{\prime }+{\cancel{\mathsf{\sigma }}}_{+}1,{\mathsf{\lambda }}_{-}^{\prime }+{\cancel{\mathsf{\sigma }}}_{-}1)}(\mathsf{\eta })\hfill \\ ={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{+}+1)(1-{\cancel{\mathsf{\sigma }}}_{+}1)(\mathsf{\eta }+1)+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{-}+1)(1-{\cancel{\mathsf{\sigma }}}_{-}1)(\mathsf{\eta }-1)\hfill \end{array}$$

(152)

Setting

$$*{\mathsf{\lambda }}_{\pm }:={\mathsf{\lambda }}_{\pm }^{\prime }+{\cancel{\mathsf{\sigma }}}_{\pm }1\equiv {\cancel{\mathsf{\sigma }}}_{\pm }({\mathsf{\lambda }}_{\pm }+1),$$

(153)

we can alternatively re-write (147) as the eigenvalue problem [15,16,17]

$$\left[{\mathbf{T}}_{{}^{\ast }\mathsf{\lambda }_{+},{}^{\ast }\mathsf{\lambda }_{-},\mathrm{m}}^{(\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}})}+{\mathsf{\epsilon }}_{\mathrm{n}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })-{\mathsf{\epsilon }}_0({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })\right]{\mathfrak{P}}_{\mathrm{m}+\mathrm{n}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{m};\mathrm{n}]=0$$

(154)

for the four exceptional differential operators

$$\begin{gathered} {\mathbf{T}}_{\mathsf{\alpha },\mathsf{\beta }}^{(\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}})}(\mathsf{\eta }):={\mathbf{T}}_{\mathsf{\alpha },\mathsf{\beta }}(\mathsf{\eta })-2(1-{\mathsf{\eta }}^2)ld {\mathrm{P}}_{\mathrm{m}}^{({\cancel{\mathsf{\sigma }}}_{+}\mathsf{\alpha }-1,{\cancel{\mathsf{\sigma }}}_{- }\mathsf{\beta }-1)}(\mathsf{\eta }){\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}} \\ -2\mathsf{\beta }(1-\mathsf{\eta }) ld {\mathrm{P}}_{\mathrm{m}}^{(-\mathsf{\alpha }-1,\mathsf{\beta }-1)}(\mathsf{\eta })+\mathrm{m}(\mathsf{\alpha }-\mathsf{\beta }-\mathrm{m}+1), \end{gathered}$$

(155)

where

$${\mathbf{T}}_{\mathsf{\alpha },\mathsf{\beta }}(\mathsf{\eta }):=(1-{\mathsf{\eta }}^2){\displaystyle \frac{{\mathrm{d}}^2 }{\mathrm{d}{\mathsf{\eta }}^2}}+2 {\mathrm{P}}_1^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta }){\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}.$$

(156)

In particular, combining (153) with (133) for $\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=+-$:

$$\mathsf{\beta }=*{\mathsf{\lambda }}_{-}={\mathsf{\lambda }}_{-}^{\prime }+1, \mathsf{\alpha }=*{\mathsf{\lambda }}_{+}={\mathsf{\lambda }}_{+}^{\prime }-1,$$

(157)

one can then directly verify that the corresponding operator (155) precisely matches (2.2) in [17]. Any polynomial from the X_m-Jacobi DPS of series J2 ($\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=+-$) is the eigenpolynomial of the operator (156) with the eigenvalue

$${\mathsf{\epsilon }}_0({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })-{\mathsf{\epsilon }}_{\mathrm{n}-\mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })=-(\mathrm{n}-\mathrm{m})(\mathsf{\alpha }+\mathsf{\beta }+\mathrm{n}-\mathrm{m}+1),$$

(158)

with n = m +j standing for the polynomial degree (in agreement with the conventional results [15,16,17] for the X_m-Jacobi orthogonal polynomials of the given series).

Let us now remind the reader that the latter operator and the Liouville potential (100) were obtained using the same RefPF (82) as the starting point and that the Liouville potential is related to the RefPF via the elementary Formula (88). This serves as the additional argument in support of the correctness of our expression for the Liouville potential of type $\tilde{\mathit{b}}$, compared with the corresponding formulas in [11,12].

The X_m-Jacobi OPS of series J2 (and therefore its counter-part in the reflected argument) have been analyzed in a more rigorous way in [15,16,17], and for this reason we focus solely on the finite EOP sequences. In Section 5 we present the universal method for constructing the finite EOP sequences formed by $\mathrm{R}\mathrm{D}\mathfrak{T}\mathrm{s}$ of the R-Jacobi polynomials. In the Section 5.1, Section 5.2 and Section 5.3 we then separately focus on the EOPs forming the eigenfunctions of the rationally extended h-PT potentials of types $\underset{˜}{\mathit{a}}$, $\underset{˜}{\mathit{b}}$, and ${\underset{˜}{\mathit{a}}}^{\prime }$.

5. Isospectral Triplet of RCSLEs Solved via $\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}\mathrm{s}$ of R-Jacobi Polynomials

Starting from this point, we discuss only the admissible RRZTs using the TFs ${\mathsf{\varphi }}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ]$ with no nodes in the interval (1, ∞) for the specified ranges of the parameters ${\mathsf{\lambda }}_{-},{\mathsf{\lambda }}_{+}$. In this Section we will consider only the TFs with the vector $\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ lying in the first three quadrants. As demonstrated in [1] using the Klein formula [49], the admissible TFs also exist for certain segments of the vector $\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ in the fourth quadrant, but this family of the finite EOP sequences lies beyond the scope of this paper.

The innovatory part of this subsection is based on our extensive finding [48] that the DBC imposed on solutions of the p-SLE unambiguously selects the PFS near the given endpoint. After converting the RCSLE (103) to its prime forms on the intervals (−1, +1) and (1, ∞), we then prove that the $\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}\mathrm{s}$ of the PFSs of the JRef CSLE (1) are themselves the PFSs of the transformed RCSLE (103).

To formulate the SLP on the interval (1, ∞) we (by analogy with the analysis) presented in Section 2.3 for the JRef CSLE) first convert the RCSLE (103) to its prime form

$$\{-{\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}(\mathsf{\eta }-1) {\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathsf{\eta }}}+\cancel{\mathfrak{q}}[\mathsf{\eta }; \stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathbf{m}]+\underset{˜}{\mathsf{\epsilon }}\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}] \} \cancel{\mathsf{\Psi }}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};\underset{˜}{\mathsf{\epsilon }}|\mathbf{m}]= 0 (\mathsf{\eta }>1)$$

(159)

and then solve it under the DBCs:

$$\underset{\mathsf{\eta }\to \mathrm{r}}{lim}\cancel{\mathsf{\Psi }}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};{\underset{˜}{\mathrm{E}}}_{\mathrm{j}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})|\mathbf{m}]=0 (\mathrm{r}=1,\infty ).$$

(160)

The zero-energy free term in the SLE (159) is related to the RefPF of the RCSLE via the generic elementary Formula (31), i.e.,

$$\cancel{\mathfrak{q}}[\mathsf{\eta }; \stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathbf{m}]=-(\mathsf{\eta }-1){\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathbf{m}]+{\displaystyle \frac{1}{4(\mathsf{\eta }-1)}} (\mathsf{\eta } >1).$$

(161)

Taking into account that

$$\underset{\mathsf{\eta }\to 1}{lim}\left[(\mathsf{\eta }-1)\cancel{\mathfrak{q}}[\mathsf{\eta }; \stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathbf{m}]\right]={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}|{\mathsf{\lambda }}_{+}+1{|}^2 >0,$$

(162)

we confirm that the ChExps of the Frobenius solutions for the pole at +1 have the same absolute value, while differing by their sign. The ExpDiff for the pole of the RCSLE (103) at infinity turned out to be energy-dependent. By analogy with (34) we find that

$$\underset{\mathsf{\eta }\to \infty }{lim}\left[\mathsf{\eta }\cancel{q}[\mathsf{\eta }; \stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathbf{m}]\right]=-\underset{\mathsf{\eta }\to \infty }{lim}\left[{\mathsf{\eta }}^2{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathbf{m}]\right]+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}=0.$$

(163)

We thus assert that the ChExps of the Frobenius solutions for the pole at ∞ are real only at negative energies and have in this case the same non-zero absolute value ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{\mathsf{\epsilon }}$ for $|\mathsf{\eta } | <1$ or ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{|\underset{˜}{\mathsf{\epsilon }}|}$ for $\mathsf{\eta } >1$, while differing by their sign.

Combining (163) with the similar limit

$$\underset{\mathsf{\eta }\to 1+}{lim}\left[(\mathsf{\eta }-1)\cancel{\mathfrak{q}} [\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathbf{m}\right]=*{\mathsf{\lambda }}_{\mathrm{o};+}^2 >0$$

(164)

for the pole at +1, we conclude that the PFSs of the p-SLE (159) near both singular endpoints are unambiguously determined by the DBCs.

By applying the RRZT with the admissible TF ${\mathsf{\varphi }}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ]$ to the eigenfunction ${\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]$ of the JRef CSLE (1) with 0 ≤ j ≤ j_max and then converting the resultant function (124) to its prime form, we come to the following expression for the $\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}$ of the (j + 1)-th eigenfunction of the p-SLE (30) solved under the DBCs (35):

$$\begin{gathered} {\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]:=\cancel{\mathsf{\Psi }}[\mathsf{\eta };\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime };-{\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })|\mathbf{m}]= \\ ={\displaystyle \frac{{\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{\mathbf{1}}]}{{\mathrm{P}}_{\mathbf{m}}^{({\cancel{\mathsf{\sigma }}}_{+}{\mathsf{\lambda }}_{+}^{\prime },{\cancel{\mathsf{\sigma }}}_{-}{\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta })}}{\mathfrak{P}}_{\mathbf{m}+\mathrm{j}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m};\mathrm{j}] (\mathsf{\eta } >1) \end{gathered}$$

(165)

with 0 ≤ j ≤ j_max and

$${\mathsf{\lambda }}_{\mp }^{\prime }=\mp {\mathsf{\lambda }}_{\mathrm{o};\mp }.$$

(166)

The q-RS (165) obeys the DBCs

$$\underset{\mathsf{\eta }\to \mathrm{r}}{lim}{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]=0 (\mathrm{r}=1,\infty )$$

(167)

iff

$${\cancel{\mathsf{\psi }}}_0[1;{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{\mathbf{1}}]=0$$

(168)

and

$${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{-}^{\prime }+{\mathsf{\lambda }}_{+}^{\prime })+\mathrm{j}<0.$$

(169)

We thus assert that the given q-RS represents an eigenfunction of the p-SLE (159) solved under the DBCs provided both conditions (168) and (169) hold We will analyze these conditions on the case-by-case basis in Section 5.1, Section 5.2 and Section 5.3 for the vector $\stackrel{\rightharpoonup }{\mathsf{\lambda }}=\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }$ lying in the first, second and third quadrant accordingly. We delay the discussion of the fourth quadrant ${\mathsf{\lambda }}_{-}>0, {\mathsf{\lambda }}_{+}<0 (\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=- -)$ for a separate study.

It has been proven in [56] that the eigenfunction of the generic SLE solved under the DBCs must be mutually orthogonal with the weight (32) on the infinite interval in question:

$${\displaystyle \underset{1}{\overset{\infty }{\int }}\mathrm{d}\mathsf{\eta }} {\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=0$$

(170)

$$\mathrm{f}\mathrm{o}\mathrm{r} 0\le {\mathrm{j}}^{\prime }<\mathrm{j}\le {\mathrm{j}}_{\max }.$$

Consequently, the polynomial components of the quasi-rational eigenfunctions (165) must be mutually orthogonal with the m-dependent weight

$$\mathrm{W} [\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]:={\displaystyle \frac{{\cancel{\mathsf{\psi }}}_0^2[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{1}]}{|{\mathrm{P}}_{\mathbf{m}}^{({\cancel{\mathsf{\sigma }}}_{+}{\mathsf{\lambda }}_{+}^{\prime },{\cancel{\mathsf{\sigma }}}_{-}{\mathsf{\lambda }}_{-}^{\prime })}(\mathsf{\eta }){|}^2}}\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}] \mathrm{f}\mathrm{o}\mathrm{r} \mathsf{\eta }>1,$$

(171)

namely,

$$\begin{array}{c}{\displaystyle \underset{1}{\overset{\infty }{\int }}\mathrm{d}\mathsf{\eta }} {\mathfrak{P}}_{\mathbf{m}+\mathrm{j}-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m};\mathrm{j}] {\mathfrak{P}}_{\mathbf{m}+{\mathrm{j}}^{\prime }-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m};{\mathrm{j}}^{\prime }] \mathrm{W} [\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]=0\hfill \\ \hfill \mathrm{f}\mathrm{o}\mathrm{r} 0\le {\mathrm{j}}^{\prime }<\mathrm{j}\le {\mathrm{j}}_{\max }\end{array}$$

(172)

Let us now demonstrate the power of the developed formalism by proving that the q-RSs (165) form the complete set of the eigenfunctions for the formulated Dirichlet problem.

Theorem 2.

The Dirichlet problem for the p-SLE (159) defined on the interval (1, ∞) with  $\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\ne --$  does not have any solutions other than the eigenfunctions (165), assuming that the constraints (168) and (169) hold and that the pole of the RCSLE (103) at +1 lies within the LP region.

Proof. 

Suppose that the given Dirichlet problem on the interval (1, ∞) has a solution ${\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]$ at an energy

$${\underset{˜}{\mathsf{\epsilon }}}_{\mathrm{n}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m})\ne -{\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }) \mathrm{for} 0\le \mathrm{j}\le {\mathrm{j}}_{\max }.$$

(173)

The RRZT of the RCSLE (103) with the TF

$$*{\mathsf{\varphi }}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]:={\displaystyle \frac{\sqrt{{\mathsf{\eta }}^2-1}}{{\mathsf{\varphi }}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}}={\displaystyle \frac{\sqrt{1+\mathsf{\eta }}}{{\cancel{\mathsf{\psi }}}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}} (\mathsf{\eta } >1)$$

(174)

converts the extraneous eigenfunction into the following q-RS of the p-SLE (30):

$$\mathsf{\Psi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]:=\sqrt{\mathsf{\eta }+1}{\displaystyle \frac{\mathrm{W}\left\{*{\mathsf{\varphi }}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}],{\mathsf{\varphi }}_{\mathrm{n}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]\right\}}{*{\mathsf{\varphi }}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}}$$

(175)

$$={\cancel{\mathsf{\psi }}}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]\mathrm{W}\left\{\sqrt{\mathsf{\eta }+1}{\cancel{\mathsf{\psi }}}_{\mathbf{m}}^{-1}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}],\sqrt{\mathsf{\eta }-1}{\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]\right\}$$

(176)

$$={\displaystyle \frac{\mathsf{\eta }+1}{{\cancel{\mathsf{\psi }}}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}}{\displaystyle \frac{\mathrm{d} }{\mathrm{d} \mathsf{\eta }}}\left\{\sqrt{{\displaystyle \frac{\mathsf{\eta }-1}{\mathsf{\eta }+1}}}{\cancel{\mathsf{\psi }}}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}] {\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]\right\} (\mathsf{\eta } >1),$$

(177)

i.e.,

$$\begin{gathered} \mathsf{\Psi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]=\sqrt{{\mathsf{\eta }}^2-1}\left\{{\stackrel{•}{\cancel{\mathsf{\psi }}}}_{\mathrm{n}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]+ ld {\cancel{\mathsf{\psi }}}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}] {\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]\right\} \\ +{\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]/\sqrt{{\mathsf{\eta }}^2-1} (\mathsf{\eta } >1), \end{gathered}$$

(178)

keeping in mind that

$$(\mathsf{\eta }+1){\displaystyle \frac{\mathrm{d} }{\mathrm{d} \mathsf{\eta }}}\sqrt{{\displaystyle \frac{\mathsf{\eta }-1}{\mathsf{\eta }+1}}}=\sqrt{{\mathsf{\eta }}^2-1} ld \sqrt{{\displaystyle \frac{\mathsf{\eta }-1}{\mathsf{\eta }+1}}}={\displaystyle \frac{1}{\sqrt{{\mathsf{\eta }}^2-1}}}.$$

(179)

Let us now recall that the extraneous eigenfunction (like any other) must decay as

$${\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]\sim {(\mathsf{\eta }-1)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} *{\mathsf{\lambda }}_{\mathrm{o};+}} \mathrm{for} 0<\mathsf{\eta }-1<<1$$

(180)

in the limit $\mathsf{\eta }\to 1+$. Examination of the sum (176) then shows that it vanishes at the lower end of the interval (1, ∞) as far as the pole the RCSLE (103) at +1 lies in the LP region (${}^{\ast }\mathsf{\lambda }_{\mathrm{o};+}>1$).

Furthermore, taking into account that

$${\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m}]\sim {\mathsf{\eta }}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{-{\underset{˜}{\mathsf{\epsilon }}}_{\mathrm{n}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime } |\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathbf{m})}}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{for} \mathsf{\eta }>>1$$

(181)

and

$$2 ld {\cancel{\mathsf{\psi }}}_{\mathbf{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}] \approx ({\mathsf{\lambda }}_{-}+{\mathsf{\lambda }}_{+}+2\mathbf{m}+1)/\mathsf{\eta } \mathrm{for} \mathsf{\eta }>>1,$$

(182)

one can verify that the solution in question also vanishes in the limit η → ∞ and therefore represents an eigenfunction of the p-SLE (30) with an eigenvalue differing from any of the eigenvalues (40) with j ≤ j_max. This result contradicts to the fact that the cited energies represent the complete discrete energy spectrum of the given Dirichlet problem. We thus confirmed that the p-SLE (159) solved under the DBCs (160) may not have eigenvalues other than (40) in the LP range of the parameter ${}^{\ast }\mathsf{\lambda }_{\mathrm{o};+}$. □

The direct consequence of the proven theorem is that the Wronskians (116) or $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}\mathrm{s}$ (119) forming the eigenfunctions (165) have exactly j + 1 real zeros larger than 1.

In the following subsections we discuss three distinguished finite EOP subsets of the X_m-DPSs of series J1, W, and J2. While each of the X-DPSs of series J1 and J2 also contain X_m-OPSs, the X_m-DPS of series W may not comprise any, because the RSLP on the interval (−1, +1) has infinitely many eigenfunctions. On other hand, the RSLP on the infinite interval has only a finite number of the eigenfunctions and as a result the sequence of Jacobi polynomials with the indexes in the second quadrant contains an infinite subsequence of Jacobi polynomials with no real zeros larger than 1.

5.1. Infinitely Many EOP Sequences of Series $\underset{˜}{\mathit{a}}$

Let us start from the vector $\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ lying in the first quadrant:

$$\stackrel{\rightharpoonup }{\mathsf{\sigma }}=\mathbf{+} \mathbf{+},\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=- \mathbf{+},$$

(183)

$${}^{\ast }\mathsf{\lambda }_{\mathrm{o};\mp }={\mathsf{\lambda }}_{\mathrm{o};\mp }+1>1$$

(184)

i.e., the ExpDiff for the pole of the CSLE (103) necessarily lies within the LP range. As the direct consequence of (183), the seed polynomial in the denominator of the weight (169) turns into the classical Jacobi polynomial with positive indexes and therefore all the poles of the RCSLE (103) are located in the closed interval [−1,+1]. Ironically, this most obvious case has never been discussed in the literature and our study of these EOP sequences in [24] did not receive a proper response.

The finite EOP sequence under consideration exists in any X_m-Jacobi DPS of series J1 specified by the upper sign in (127), i.e., $\mathbf{m}$ is allowed to take any positive value m. Bearing in mind (166), we can represent this X-orthogonal polynomial subset as

$$\begin{gathered} {\mathfrak{P}}_{\mathrm{m}+{\mathrm{j}}^{\prime }}[\mathsf{\eta };-{\mathsf{\lambda }}_{\mathrm{o};-},{\mathsf{\lambda }}_{\mathrm{o};+}|- +,\mathrm{m};\mathrm{j}]=(\mathsf{\eta }+1){\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{\mathrm{o};+}, {\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta }){\stackrel{•}{\mathrm{P}}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{\mathrm{o};+}, -{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta }) \\ -({\mathsf{\lambda }}_{\mathrm{o};-}+\mathrm{m}){\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{\mathrm{o};+}+1,{\mathsf{\lambda }}_{\mathrm{o};-}-1)}(\mathsf{\eta }){\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{\mathrm{o};+}, -{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta }) \end{gathered}$$

(185)

$$\begin{array}{c}\hfill \mathrm{for}0\le \mathrm{j}\le {\mathrm{j}}_{\max }.\end{array}$$

Setting z = −$\mathsf{\eta }$, $\mathsf{\alpha }={\mathsf{\lambda }}_{\mathrm{o};-}, \mathsf{\beta }= {\mathsf{\lambda }}_{\mathrm{o};+}$, we come to the polynomials (72) in [16]:

$$\begin{gathered} {(-1)}^{\mathrm{j}+\mathrm{m}+1}{\mathfrak{P}}_{\mathrm{j}+\mathrm{m}}[-\mathrm{z};-{\mathsf{\lambda }}_{\mathrm{o};-},{\mathsf{\lambda }}_{\mathrm{o};+}|- +,\mathrm{m};\mathrm{j}]=(\mathsf{\alpha }+\mathrm{m}){\mathrm{P}}_{\mathrm{m}}^{(\mathsf{\alpha }-1,\mathsf{\beta }+1)}(\mathrm{z}){\mathrm{P}}_{\mathrm{j}}^{(-\mathsf{\alpha }, \mathsf{\beta })}(\mathrm{z}) \\ -(\mathrm{z}-1){\mathrm{P}}_{\mathrm{m}}^{(\mathsf{\alpha },\mathsf{\beta })}(\mathrm{z}){\displaystyle \frac{\mathrm{d} }{\mathrm{d}\mathrm{z}}}{\mathrm{P}}_{\mathrm{j}}^{( -\mathsf{\alpha },\mathsf{\beta })}(\mathrm{z}), \end{gathered}$$

(186)

with the interchanged degrees m and j, i.e.,

$${(-1)}^{\mathrm{m}+1}{\mathfrak{P}}_{\mathrm{m}+\mathrm{j}}[-\mathrm{z};-\mathsf{\alpha },\mathsf{\beta }|- +,\mathrm{m};\mathrm{j}]=(\mathsf{\alpha }+\mathrm{m}){\stackrel{⌢}{P}}_{\mathrm{j},\mathrm{j}+\mathrm{m}}^{(\mathsf{\alpha }-1,\mathsf{\beta }+1)}(\mathrm{z})$$

(187)

$$\mathrm{for} 0\le \mathrm{j}\le {\mathrm{j}}_{\max }=⌊{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}(\mathsf{\alpha }-\mathsf{\beta }-1)⌋.$$

We thus proved that the polynomial (183) converted to its monic form coincides with the monic (j + m)-th degree polynomial from the X_j-Jacobi OPS of series J1 [24]. This implies that the polynomials (187) arranged into the infinite row (m = 1, 2, …) at fixed j are mutually orthogonal with the weight

$${\mathrm{W}}_{{\mathsf{\lambda }}_{\mathrm{o};-},{\mathsf{\lambda }}_{\mathrm{o};+}} [\mathsf{\eta };{}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}}|-+,\mathrm{j} ]:={\displaystyle \frac{{(1+\mathsf{\eta })}^{{\mathsf{\lambda }}_{\mathrm{o};-}} {(1-\mathsf{\eta })}^{{\mathsf{\lambda }}_{\mathrm{o};+}}}{|{\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{\mathrm{o};+}-1,-{\mathsf{\lambda }}_{\mathrm{o};-}-1)}(\mathsf{\eta }){|}^2}} \mathrm{for}0\le \mathrm{j}<{\mathrm{j}}_{\max }$$

(188)

(cf. (87) in [16]). This brings us to the (j_max+ 1) $\times$ ∞ rectilinear polynomial matrix mentioned in Introduction.

Making use of (183) and (184), one finds

$${\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{\mathbf{1}}=\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times ({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}+\stackrel{\rightharpoonup }{\mathbf{1}})=\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times {}^{\ast }\stackrel{\rightharpoonup }{\mathsf{\lambda }}_{\mathrm{o}},$$

(189)

so the q-RS (165) takes the form:

$$\begin{gathered} {\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };-{\mathsf{\lambda }}_{\mathrm{o};-},{\mathsf{\lambda }}_{\mathrm{o};+}|- +,\mathrm{m}]= \\ ={\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };-{}^{\ast }\mathsf{\lambda }_{\mathrm{o};-},{}^{\ast }\mathsf{\lambda }_{\mathrm{o};+}]{\displaystyle \frac{{\mathfrak{P}}_{\mathrm{m}+\mathrm{j}}[\mathsf{\eta };-{\mathsf{\lambda }}_{\mathrm{o};-},{\mathsf{\lambda }}_{\mathrm{o};+}|- +,\mathrm{m};\mathrm{j}]}{{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{\mathrm{o};+},{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta })}}(\mathsf{\eta } >1). \end{gathered}$$

(190)

Keeping in mind that

$${}^{\ast }\mathsf{\lambda }_{\mathrm{o};-}-{}^{\ast }\mathsf{\lambda }_{\mathrm{o};+}={\mathsf{\lambda }}_{\mathrm{o};-}-{\mathsf{\lambda }}_{\mathrm{o};+} >2\mathrm{j}+1,$$

(191)

we conclude that that the function ${\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };-{}^{\ast }\mathsf{\lambda }_{\mathrm{o};-},{}^{\ast }\mathsf{\lambda }_{\mathrm{o};+}]$ is the eigenfunction of the p-SLE (30), satisfying by definition the DBC (36). (As expected, the absolute values of the power exponents $\mp {\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}*{\mathsf{\lambda }}_{\mathrm{o};\mp }$ of η $\pm 1$ coincide with halves of the ExpDiffs $*{\mathsf{\lambda }}_{\mathrm{o};\mp }$ for the poles of the RCSLE (103) at $\mp$1). The condition (169) also holds since

$${\mathsf{\lambda }}_{-}^{\prime }+{\mathsf{\lambda }}_{+}^{\prime }={\mathsf{\lambda }}_{\mathrm{o};+}-{\mathsf{\lambda }}_{\mathrm{o};-}<-2\mathrm{j}-1.$$

(192)

This confirms that the q-RSs (190) are the eigenfunctions of the p-SLE (159) solved under the DBCs (160) and therefore their polynomial components are mutually orthogonal with the weight

$$\mathrm{W} [\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|- +,\mathrm{m}]={\displaystyle \frac{{\cancel{\mathsf{\psi }}}_0^2[\mathsf{\eta };-{\mathsf{\lambda }}_{\mathrm{o};-},{}^{\ast }\mathsf{\lambda }_{\mathrm{o};+}]}{|{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{\mathrm{o};+},-{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta }){|}^2}} (\mathsf{\eta } >1).$$

(193)

Since all the prerequisites for Theorem 2 are valid, the analytical Formula (40) with j ≤ j_max determines all the eigenvalues of the p-SLE (159) solved under the DBCs is exactly solvable.

From the perspective of the quantum mechanical applications, the corresponding isospectral Liouville potential (100) with 1 < $*{\mathsf{\lambda }}_{\mathrm{o};+}$ < 2 constitutes the very special case, because its SUSY partner, h-PT potential with the positive parameter ${\mathsf{\lambda }}_{\mathrm{o};+}$ smaller than 1 (¹/₂ < $\mathrm{B}-\mathrm{A}$ < ³/₂) has only the CDBESs. This implies that the discrete energy spectrum for the Liouville potential (100) with 1 < $*{\mathsf{\lambda }}_{\mathrm{o};+}$ < 2 cannot be found using the conventional rules of the SUSY quantum mechanics.

5.2. Infinitely Many EOP Sequences of Series $\underset{˜}{{\mathit{a}}^{\prime }}$

Let us now consider the peculiar case when both vector $\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ and ${\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }$ lie in the same (second) quadrant ($\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=+ +$). The admissible TFs of this type exist only for finite EOP sequences. According to the inequality (65), the q-RS

$${\cancel{\mathsf{\psi }}}_{\underset{˜}{{\mathit{a}}^{\prime }}, \mathbf{m}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]={\displaystyle \frac{1}{\sqrt{\mathsf{\eta }-1}}}{\mathsf{\varphi }}_{++, \mathbf{m},\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}] (\mathsf{\eta } >1)$$

(194)

is admissible for any $\mathbf{m}$ from the infinite sequence of the positive integers starting from $⌈{\mathsf{\lambda }}_{\mathrm{o};-}-{\mathsf{\lambda }}_{\mathrm{o};+}-1⌉$. Substituting (115) into (123) with $\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ = ${\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }$ gives

$${\cancel{\mathsf{\psi }}}_{\underset{˜}{{\mathit{a}}^{\prime }}, \mathbf{m}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]= {\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{\mathbf{1}}]{\displaystyle \frac{{\mathrm{W}}_{\mathbf{m}+\mathrm{j}-1}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}[\mathsf{\eta }|\mathbf{m},\mathrm{j}]}{{\mathrm{P}}_{\mathbf{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })}} (\mathsf{\eta } >1).$$

(195)

As expected, the absolute values of the power exponents ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mp }+1)$ of η $\pm 1$ coincide with halves of the ExpDiffs $*{\mathsf{\lambda }}_{\mathrm{o};\mp }=|{\mathsf{\lambda }}_{\mp }\mp 1|$ for the poles of the RCSLE (103) at $\mp 1$. Since ${\cancel{\mathsf{\sigma }}}_{+}=+$, ${\mathsf{\lambda }}_{+}={\mathsf{\lambda }}_{+}^{\prime }={\mathsf{\lambda }}_{\mathrm{o};+}$, and therefore

$${\mathsf{\lambda }}_{+}^{\prime }+1={}^{\ast }\mathsf{\lambda }_{\mathrm{o};+}={\mathsf{\lambda }}_{\mathrm{o};+}+1,$$

we confirm that the q-RS (195) necessarily vanishes at +1.

To confirm the DBCs (167) at infinity, first note that the PF in the right-hand side of (195) grows at large η as η^j−1. Taking into account that the q-RSs ${\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]$ and ${\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{\mathbf{1}}]$ have the same asymptotics at infinity, while the eigenfunction ${\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]$ with j ≤ j_max satisfies the DBC at this endpoint, we conclude that the q-RS (195) vanishes at infinity, as required.

Since $*{\mathsf{\lambda }}_{\mathrm{o};+}>1$, Theorem 2 again assures the exact solvability of the Dirichlet problem in question. Once more, the corresponding Liouville potential (100) with 1 < $*{\mathsf{\lambda }}_{\mathrm{o};+}$ < 2 and $\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=+ +$ is not covered by the conventional rules of the SUSY quantum mechanics. In Grandati’s notation [9]: ${\mathsf{\lambda }}_{\mathrm{o};+}=\mathsf{\alpha }, {\mathsf{\lambda }}_{\mathrm{o};-}=\mathsf{\beta }$ so his analysis is applicable solely to the parameter range $\mathsf{\beta }>\mathsf{\alpha }+1>2.$ On the contrary, the technique developed here made it possible to extend the discussion of the rationally-extended h-PT potentials to the parameter range $0<\mathsf{\alpha }<1.$ The border case $\mathsf{\alpha }=1$ requires a more cautious analysis.

I doubt that the extension of the domain definition for the parameter α to negative values (α > −${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ in (31) in [9]) makes any sense in the quantum-mechanical studies of the h-PT potential and its rational extensions. Though it is possible that formulating the appropriate PSLPs would allow one to construct finite EOP sequences of types $\underset{˜}{\mathit{a}}$ and $\underset{˜}{{\mathit{a}}^{\prime }}$ for −1 < α < 0 the q-RSs composed of these polynomials could be hardly applied in physics.

5.3. Finitely Many EOP Sequences of Series $\underset{˜}{\mathit{b}}$

By placing the vector $\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ into the third quadrant, we finally come to the case which so far has attracted much more attention from the physicists [10,11,12,13]—the rationally extended h-PT potential of type $\underset{˜}{\mathit{b}}$ constructed using the finite number of the nodeless TFs ${\mathsf{\varphi }}_{\mathbf{m}}[\mathsf{\eta };-{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}} ]$ under the constraint (63). In the sharp contrast with the two cases discussed above, the RRZTs in question decreases by 1 the ExpDiff for the pole at +1. As a result, the p-SLE (159) can be analytically solved within the LC range of the ExpDiff ${\mathsf{\lambda }}_{\mathrm{o};+}$. However, we were unable to prove that the SLP is exactly solvable in this case. The most serious setback of this failure is that our formalism cannot anymore guarantee that the $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}\mathrm{s}$ (119) forming the eigenfunctions (165) with $\stackrel{\rightharpoonup }{\mathsf{\lambda }}=-{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}$ and $\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=+ -,$ have exactly j + 1 real zeros larger than 1 if ${\mathsf{\lambda }}_{\mathrm{o};+}$ < 1.

Keeping in mind that

$$\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{\mathbf{1}}=\stackrel{\rightharpoonup }{\mathbf{1}}-{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}$$

(196)

and

$${\mathsf{\lambda }}_{\mp }^{\prime }\pm 1=\mp ({\mathsf{\lambda }}_{\mathrm{o};\mp }-1)=\pm ({\mathsf{\lambda }}_{\mp }+1),$$

(197)

one finds

$${}^{\ast }\mathsf{\lambda }_{\mathrm{o};\mp }= |{\mathsf{\lambda }}_{\mathrm{o};\mp }-1| = |{\mathsf{\lambda }}_{\mp }^{\prime }\pm 1|.$$

(198)

As expected, the absolute values of the power exponents $\mp$¹/₂$*{\mathsf{\lambda }}_{\mathrm{o};\mp }$ of η$\pm 1$ in (165) thus coincide with halves of the ExpDiffs for the poles of the RCSLE (103) at $\mp 1$, provided that the latter lay in the LP region.

Since

$${\mathsf{\lambda }}_{+}^{\prime }+1={\mathsf{\lambda }}_{\mathrm{o};\mp }-1 >0 \mathrm{i}\mathrm{f}\mathrm{f}{\mathsf{\lambda }}_{\mathrm{o};+}>1,$$

(199)

the q-RS (165) obeys the DBC at +1 iff the pole of the CSLE (1) at this end is LP.

Furthermore,

$${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{-}^{\prime }+{\mathsf{\lambda }}_{+}^{\prime })+\mathrm{j}={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};+}-{\mathsf{\lambda }}_{\mathrm{o};-}+2\mathrm{j})<0,$$

(200)

so the second prerequisite (169) for Theorem 2 also holds.

It is interesting to mention that the basic PFS of the given type ($\mathbf{m}$ = 0) have been used in [84] to study the DTs between the LP and LC regions of the h-PT potential. Keeping in mind that

$$\mathrm{A}:={\mathsf{\lambda }}_{\mathrm{o};-}={}^{\ast }\mathsf{\lambda }_{\mathrm{o};-}-1, \mathrm{B}={\mathsf{\lambda }}_{\mathrm{o};+}-1={}^{\ast }\mathsf{\lambda }_{\mathrm{o};+}<1$$

in our terms, we find the close connection between the analysis presented by Gangopadhyaya et al. [84] and the current discussion. Namely, the cited authors pick up the $\mathrm{D}\mathfrak{T}$ of each eigenfunction (well-defined in the LP region) as the special representative of the CDBESs for the eigenvalue. As more recently proven in [48], the $\mathrm{R}\mathrm{R}\mathrm{Z}\mathfrak{T}$ of the eigenfunction of the p-SLE with the LP singularities at the ends necessarily obeys the DBCs under consideration. In other words, the cited authors select the bound energy states described by the PFSs at the origin, with a clear resemblance to our prescriptions.

6. Discussion

The paper is based on the three core notions advanced by the author in the aforementioned publications. One of them is the scrupulous analysis of the RCSLEs obtained by the RRZTs of the JRef CSLE (1). As pointed to in Introduction, any RRZT of the RCSLE is directly related to a DT of the corresponding Liouville potential and as a result this part of our studies lays the rigorous foundation for the SUSY theory of the quantum-mechanical potentials solvable by polynomials.

Another important element of our approach is the concept of the ‘prime’ SLE (p-SLE) chosen in such a way that the two ChExps for the poles at the endpoints differ only by sign. As a result, the energy spectrum of the given Sturm-Liouville problem can be obtained by solving the p-SLE under the DBCs. This in turn allows one to take advantage of the rigorous theorems proven in [56] for eigenfunctions of the generic SLE solved under the DBCs.

Finally, we put forward the concept of the X-Jacobi DPSs formed by polynomial solutions of the Bochner-type ODEs. Since the X-DPSs either do not start from a constant or lack a first-degree polynomial, they do not satisfy the Bochner theorem [36], as originally noticed by Gómez-Ullate et al. [33] in the context of the discovered by them X₁-Jacobi OPS. In general, each X-Jacobi OPS belongs to one of the X-Jacobi DPSs. If the given X-Jacobi DPS does contain a X-Jacobi OPS, then we say that they belong to the same series. This is why we [8] started referring to the X_m-Jacobi DPS of series D [24] as being of series J3, after becoming aware of Grandati and Bérard’s [50] discovery of the X_m-Jacobi OPS of series J3 for even m. (We still continue to refer the X_m-Jacobi DPS of series D as being of series J3 for odd m even though the latter do not contain any OPSs). (As proven in [28], the X₁-Jacobi DPS of series D does not contain any finite EOP sequences either).

In addition to the two finite EOP sequences identified in [9] and in [10,11] accordingly, our original research [24] discovered the (undeniably new to our knowledge) finite EOP sequences of type $\underset{˜}{\mathit{a}}$, which were constructed using the q-RTFs composed of the classical Jacobi polynomials. Since all the zeros of the latter polynomials lie between −1 and +1, the q-RTFs of this kind (representing the PFSs at the origin) do not have nodes within the quantization interval (+1, ∞). The very remarkable feature of the finite EOP sequences constructed in such a way [24] is that they can be arranged into the rectilinear polynomial matrix with a finite number of rows and an infinite number of columns representing the X-Jacobi OPSs of series J1 and $\mathrm{R}\mathrm{D}\mathfrak{T}\mathrm{s}$ of the R-Jacobi polynomials accordingly.

Compared with the cited preprints [1,8,24], the brand new element of the current analysis, is the proof that the X_m-Jacobi DPSs of series J1 and J2 are composed of the $\mathrm{p}\text{-}\mathrm{W}\mathrm{s}$ of two Jacobi polynomials [51], and therefore this is also true for their infinite and finite X-orthogonal subsets. In particular the mentioned rectilinear polynomial matrix is formed by the $\mathrm{p}\text{-}\mathrm{W}\mathrm{s}$ of the classical Jacobi and R-Jacobi polynomials, respectively.

The simple $\mathrm{p}\text{-}\mathrm{W}\mathrm{P}$ representation of the X-orthogonal subsets of the X_m-Jacobi DPS of series J1 has very interesting far-reaching implications, making it possible to obtain analytical expressions for both infinite and finite $\mathrm{R}\mathrm{D}\mathrm{C}\mathfrak{T}\mathrm{s}$ of the classical Jacobi and R-Jacobi polynomials using the PFSs of the same type $\underset{˜}{\mathit{a}}$ as seed functions. In [25] we have discussed the RCSLEs obtained from the JRef CSLE (1) using the seed Jacobi polynomials with the same indexes. Since the RDCTs using the seed functions of types $\underset{˜}{\mathit{a}}$ and $\underset{˜}{\mathit{b}}$ are specified by same series of the Maya diagrams [88], any RCSLE using an arbitrary combination of these seed functions can be alternatively obtained by considering only infinitely many combinations ${\overline{\mathrm{M}}}_{\mathrm{p}}:=\{{\mathrm{m}}_1,{\mathrm{m}}_2,\dots ,{\mathrm{m}}_{\mathrm{p}}\}$ of the PFSs of the same type $\underset{˜}{\mathit{a}}$ [51,72,89]. In particular, the Liouville potentials constructed by means of the TFs $\underset{˜}{\mathit{b}},\mathrm{m}$ discussed above can be alternatively obtained by means of the $\mathrm{m}$^th-order RDCT with the seed functions $\underset{˜}{\mathit{a}}, \tilde{\mathrm{m}}=1,\dots ,\mathrm{m}$ [72,89].

One can easily verify that the quasi-rational functions defined via the relations

$$\begin{gathered} {\mathfrak{P}}_{\mathfrak{N}({\overline{\mathrm{M}}}_{\mathrm{p}})+\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime } |- +⋮{\overline{\mathrm{M}}}_{\mathrm{p}};\mathrm{j}]:={\displaystyle \prod _{\aleph =\mp }(}\mathsf{\eta }-\aleph 1{)}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{p}({\mathsf{\lambda }}_{\aleph }+1)-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\aleph }^{\prime }+1)} \\ \times {(\mathsf{\eta }+1)}^{-\mathrm{p}} \mathrm{W}\{{\mathsf{\varphi }}_{{\mathrm{m}}_1}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ]\dots ,{\mathsf{\varphi }}_{{\mathrm{m}}_{\mathrm{p}}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ],{\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime } ]\}, \end{gathered}$$

(201)

where ${\mathsf{\lambda }}_{\mp }^{\prime }=\mp {\mathsf{\lambda }}_{\mp }$, constitute polynomials of degree $\mathfrak{N}({\overline{\mathrm{M}}}_{\mathrm{p}})+\mathrm{j}$, where [90]

$$\mathfrak{N}({\overline{\mathrm{M}}}_{\mathrm{p}})={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{p}}{\mathrm{m}}_{\mathrm{k}}}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{p}(\mathrm{p}-1).$$

(202)

The explicit expression for the ‘simple’ $\mathrm{p}\text{-}\mathrm{W}$ polynomials (201) can be easily obtained using (92) in [51] and will be examined in a separate paper. As expected, the ‘simple’ $\mathrm{p}\text{-}\mathrm{W}$ polynomials (201) turn into the Xm-Jacobi DPSs of series J1 for p = 1.

It has been proven in [25] that the ExpDiffs for the poles of the RCSLE

$$\left\{{\displaystyle \frac{{\mathrm{d}}^2 }{\mathrm{d}{\mathsf{\eta }}^2}}+{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|{\overline{\mathrm{M}}}_{\mathrm{p}}]+\underset{˜}{\mathsf{\epsilon }} \mathit{\rho } [\mathsf{\eta }] \right\} \mathsf{\Phi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};\underset{˜}{\mathsf{\epsilon }}|{\overline{\mathrm{M}}}_{\mathrm{p}}]= 0$$

(203)

at $\mp 1$ are given by the simple formula

$$*{\mathsf{\lambda }}_{\mathrm{o};\mp }^{(\mathrm{p})}=|{\mathsf{\lambda }}_{\mp }+\mathrm{p}|.$$

(204)

The corresponding eigenfunctions have the form [91]:

$$\mathsf{\Phi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};{\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })|+ +⋮{\overline{\mathrm{M}}}_{\mathrm{p}};- +,\mathrm{j}]={\displaystyle \frac{\mathrm{W}\{{\mathsf{\varphi }}_{{\mathrm{m}}_1}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ]\dots ,{\mathsf{\varphi }}_{{\mathrm{m}}_{\mathrm{p}}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ],{\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime } ]\}}{{\mathit{\rho }}^{\raisebox{1ex}{$\mathrm{p}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} [\mathsf{\eta }]\mathrm{W}\{{\mathsf{\varphi }}_{{\mathrm{m}}_1}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ]\dots ,{\mathsf{\varphi }}_{{\mathrm{m}}_{\mathrm{p}}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} ]\} }}$$

(205)

$$={\left({\displaystyle \frac{\mathsf{\eta }-1}{\mathsf{\eta }+1}}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{p}}{\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime } ]{\displaystyle \frac{{\mathfrak{P}}_{\mathfrak{N}({\overline{\mathrm{M}}}_{\mathrm{p}})+\mathrm{j}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }} |- +⋮{\overline{\mathrm{M}}}_{\mathrm{p}};\mathrm{j}]}{{\mathrm{W}}_{\mathfrak{N}({\overline{\mathrm{M}}}_{\mathrm{p}})}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}[\mathsf{\eta }|{\overline{\mathrm{M}}}_{\mathrm{p}}] }},$$

(206)

where

$${\mathrm{W}}_{\mathfrak{N}({\overline{\mathrm{M}}}_{\mathrm{p}})}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}[\mathsf{\eta }|{\overline{\mathrm{M}}}_{\mathrm{p}}]:=\mathrm{W}\{{\mathrm{P}}_{{\mathrm{m}}_1}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta }),\dots ,{\mathrm{P}}_{{\mathrm{m}}_{\mathrm{p}}}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}(\mathsf{\eta })\}.$$

(207)

Representing (206) as

$$\begin{gathered} \mathsf{\Phi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }};{\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })|+ +⋮{\overline{\mathrm{M}}}_{\mathrm{p}};- +,\mathrm{j}] \\ ={\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };-{\mathsf{\lambda }}_{\mathrm{o};-}-\mathrm{p}; {\mathsf{\lambda }}_{\mathrm{o};+}+\mathrm{p} ]{\displaystyle \frac{{\mathfrak{P}}_{\mathfrak{N}({\overline{\mathrm{M}}}_{\mathrm{p}})+\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime } |- +⋮{\overline{\mathrm{M}}}_{\mathrm{p}};\mathrm{j}]}{{\mathrm{W}}_{\mathfrak{N}({\overline{\mathrm{M}}}_{\mathrm{p}})}^{({\mathsf{\lambda }}_{+}, {\mathsf{\lambda }}_{-})}[\mathsf{\eta }|{\overline{\mathrm{M}}}_{\mathrm{p}}] }}, \end{gathered}$$

(208)

we find that the absolute values of the power exponents of η ± 1 coincide with halves of the ExpDiffs (204) and therefore the simple $\mathrm{p}\text{-}\mathrm{W}$ polynomials (194) form a X-Jacobi DPS, which will be referred to by us as being of series $- +⋮{\overline{\mathrm{M}}}_{\mathrm{p}}$. A similar X-Jacobi DPS of series $+-⋮{\overline{\mathrm{M}}}_{{\mathrm{p}}^{\prime }}$ simply constitutes another representation of one of the X-DPSs mentioned above and might be dropped though it would come with a catch: the dropped net of the X-Jacobi OPSs ($\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=+ -$,${\mathsf{\lambda }}_{\mp }^{\prime }={\mathsf{\lambda }}_{\mathrm{o};\mp }^{\prime }$) starts from the infinite manifold of the X_m-Jacobi OPSs in the conventional sense [15,16,17].

If we choose ${\mathsf{\lambda }}_{\mp }={\mathsf{\lambda }}_{\mathrm{o};\mp }$ and restrict j from above by the constraint j ≤ j_max, then the resultant $\mathrm{D}\mathrm{C}\mathfrak{T}\mathrm{s}$ of the R-Jacobi polynomials form the finite EOP sequences of series $- +⋮{\overline{\mathbf{M}}}_{\mathrm{p}}$, where ${\overline{\mathbf{M}}}_{\mathrm{p}}={\mathbf{m}}_1,\dots ,{\mathbf{m}}_{\mathrm{p}}$ are the the admissible sets of the JS solutions assuring that the corresponding Jacobi polynomial Wronskians (JPWs) do not have nodes within the given quantization interval for the specified ranges of the parameters ${\mathsf{\lambda }}_{-},{\mathsf{\lambda }}_{+}$. We thus come to the very broad brand families of both infinite and finite EOP sequences, which will be examined in detail in a separate publication.

One can also combine the nodeless PFSs of type ${\underset{˜}{\mathit{a}}}^{\prime }$ with the juxtaposed pairs of the eigenfunctions (type $\underset{˜}{\mathit{c}}$) to construct the RCSLEs quantized by the finite EOP sequences composed of the Wronskian transforms of the R-Jacobi polynomials, similar to the finite EOP sequences formed by the Wronskian transforms of the R-Bessel and R-Jacobi polynomials [26,28].

The general case using the seed functions of all the four types $\underset{˜}{\mathit{a}}$, $\underset{˜}{\mathit{b}}$, ${\underset{˜}{\mathit{a}}}^{\prime }$, and $\underset{˜}{\mathit{c}}$ represent a much more challenging problem.