X₁-Jacobi Differential Polynomial Systems and Related Double-Step Shape-Invariant Liouville Potentials Solvable by Exceptional Orthogonal Polynomials

Abstract

This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak–Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux–Crum transformation. The associated Sturm–Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X₁-Jacobi DPSs composed of Heun polynomials contains both the X₁-Jacobi orthogonal polynomial system (OPS) and the finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski–Jacobi (R-Jacobi) polynomials, while the second analytically solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way.

1. Introduction

In our recently published article [1], we proved that the radial potential, exactly solvable in terms of hypergeometric functions [2], preserves its form under two sequential rational Darboux transformations (RDTs). It is essential that the first of these RDTs (using the ‘basic’ quasi-rational [3] solution (q-RS) as its transformation function (TF)) converts the Schrödinger equation with the cited potential into the new radial Schrödinger equation exactly quantized in terms of polynomial solutions of the Heun equation [4] with degree-dependent exponent parameters. It was then proven that the new radial potential also preserves its form under two sequential RDTs. It was explicitly demonstrated that the conventional rules of the SUSY quantum mechanics [5] fail if the first RDT places the centrifugal barrier into the limit-circle (LC) range, while the second transformation keeps it within the same range.

The purpose of this analysis is to show that similar results can be formulated for the rational Darboux transforms $(\mathrm{R}\mathrm{D}\text{T}\mathrm{s})$ of the trigonometric and hyperbolic Pöschl–Teller (t- and h-PT) potentials [6] quantized in terms of either infinite [7,8] or finite [9] exceptional orthogonal polynomial (EOP) solutions of the Heun equation with degree-independent exponent parameters. Note that the mentioned Heun-reference (HRef) potentials were incorrectly referred to in [10] as ‘shape-invariant’. Indeed, the RDT using the lowest-energy eigenfunction as its TF creates a new pole on the real axis and therefore necessarily changes the form of the corresponding canonical Sturm–Liouville equation (CSLE). On the other hand, as thoroughly discussed below, the Liouville potentials associated with the HRef CSLEs in question do preserve their form under two sequential (specially chosen) RDTs.

The important novel element of our approach [11,12,13] to the theory of the EOPs is that we define them as solutions of the Bochner-type [14] ordinary differential equations (ODEs) with polynomial coefficients, rather than eigenpolynomials of the rational differential operator [15,16]. The ODEs in question are constructed by applying energy-independent gauge transformations to rational CSLEs (RCSLEs). The EOPs are then defined as polynomial components of the quasi-rational eigenfunctions of the associated Sturm–Liouville problems (SLPs).

In this connection, let us remind the reader that, as stressed by Kwon and Littlewood [17], 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. This observation brought the author [13,18] to the concept of complex exceptional Bochner (X-Bochner) polynomials, which satisfy a second-order ODE of Bochner type but violate his theorem because each sequence either does not start from a constant or lacks the first-degree polynomial. We refer to these sequences as complex exceptional differential polynomial systems (X-DPSs) with the term ‘DPS’ used in exactly the same sense it is used by Everitt et al. [19,20] for conventional sequences of polynomials obeying the Bochner theorem.

It was proven by Kwon and Littlejohn [17] that all the real field reductions in the complex DPSs constitute quasi-definite orthogonal polynomial sequences, [21] and, for this reason, these authors refer to the latter as ‘OPSs’. However, this is not true for the X-DPSs, and we thus preserve the term ‘X-OPS’ solely for the sequences formed by positively definite orthogonal polynomials.

Any Bochner-type ODE with polynomial coefficients can be re-written in the form of the rational differential eigenequation, which brings us to the exceptional operator introduced in [22], with the corresponding eigenpolynomials forming a X-DPS in our terms. While the authors of the cited paper (see also their more recent work [23]) are interested solely in X-OPSs, we [11,12,18] have been developing the alternative technique covering both infinite and finite EOP sequences.

The formalism utilized by us for constructing X_m-Jacobi DPSs [11] draws on the pioneering paper by Rudyak and Zakhariev [24] introducing the so-called [25] ‘generalized Darboux transformations’ of the generic CSLE. In [26], we suggested terming them as the ‘Liouville-Darboux transformation’ (LDT) to stress that they can be re-interpreted as the three-step operations:

(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 in variable at Step (i),

making it easier to relate our results to the conventional recipes of the SUSY quantum mechanics [5]. It should be, however, stressed that the definition of these transformations [24,25,26] requires no reference to the Liouville transformation, and, for this reason, we prefer to refer to them as ‘Rudyak–Zakhariev transformations’ (RZTs) until we explicitly require for the transformation function (TF) of the given RZT not to have nodes inside the given quantization interval.

In the case of our interest, the RZT using a quasi-rational TF is applied to the Jacobi-reference (JRef) CSLE defined via (1)–(3) below. The transformed CSLE is then converted to the Bochner-type ODE with polynomial coeffiecients taking advantage of the fact that the density function of our interest has only simple poles in the finite plane, and, as a result, the mentioned gauge transformation is energy-independent [27]. Consequently, the linear coefficient function of the resultant differential equation does not depend on degrees of the sought-for polynomials.

To construct the infinite and finite EOP sequences, we take advantage of the concept of the prime RSLEs (p-RSLEs) suggested by the author [28] earlier. The gauge transformation used to convert the given RCSLE to its prime form is chosen in such a way that the characteristic exponents (ChExps) of the two Frobenius solutions near the given singular endpoint differ only by their sign. As a result, solving the resultant p-RSLE under the Dirichlet boundary conditions (DBCs) unambiguously determines the principal Frobenius solution (PFS). One of the benefits of our approach is that we can take advantage of the powerful spectral theorems proven by Gesztezy et al. [29] for solutions of the generic SLEs solved under the DBCs. In particular, we assert that the q-RSs satisfying the DBCs are necessarily orthogonal, and, therefore, this must also be true for their polynomial components.

It was revealed that the rational RZTs (RRZTs) of the JRef CSLEs with the basic TFs result in the two distinguished HRef CSLEs with the third pole lying either inside the finite interval (−1,+1) or in the negative interval (−∞,−1). In the latter case, the DBCs can be imposed at the endpoints of both intervals [−1,+1] or [1,∞]; however, it was found that the Liouville potential on the positive infinite interval does not have the discrete energy spectrum. While the SLP on the finite interval gives rise to the renowned X₁-Jacobi OPS, the eigenfunctions for the HRef CSLE with the pole between −1 and +1 are expressible in terms of RDS of the Romanovski–Jacobi (R-Jacobi) polynomials [30,31,32,33,34,35,36].

The analysis presented below represents the simplest extension of the well-known fact that the Jacobi DPS formed by Jacobi polynomials with arbitrarily chosen real indexes contains both an infinite OPS composed of classical Jacobi polynomials and a finite orthogonal sequence of the R-Jacobi polynomials. The finite EOP sequence composed of the RDS of the R-Jacobi polynomials belongs to the same X-DPS as the X₁-Jacobi OPS, and, as a result, it is formally defined by the formulas discovered in [15] for its infinite X-orthogonal reduction. Based on this observation, Yadav et al. [37] refer to the X-DPS in question simply as ‘X₁-Jacobi polynomials’. This important distinction between the X₁-Jacobi polynomials and its infinite orthogonal reduction represented by the X₁-Jacobi OPS was clarified in our works [12,38].

The SLPs sketched above can be converted by the Liouville transformation to the eigenproblems for the 1D Schrödinger equation giving rise to the two double-step shape-invariant potentials discovered by Quesne [7,8] and Bagchi et al. [9], respectively. It should be stressed that both potentials were historically constructed using the conventional SUSY theory, and the main purpose of this work is to re-formulate the cited results as some by-products of the more rigorous mathematical theory of the RSLEs solved under the DBCs [29]. The new approach developed in our works [12,38] was either overlooked or simply disregarded by Soltész et al. [10], whose argumentation is to a certain extent is reminiscent of our earlier work [27], where we made an attempt to incorporate the SUSY suppositions into the theory of RSLEs with energy-independent exponent differences (ExpDiffs) for the poles in the finite plane. Later, we realized that Quesne’s approach [7,8] to the theory of rationally extended potentials is nothing but the quantum-mechanical application of the self-contained mathematical formalism based on the RRZTs of the CSLEs solvable by the classical Jacobi and Laguerre polynomials. Since the singular Sturm–Liouville theory reached a much higher level of development, compared with the theory of the Riccati equation (which lays the foundation for the SUSY quantum mechanics), there is no advantage in re-formulating the problem in terms of the rational superpotentials as recently performed in [10].

Another spinoff advancement initiated in this paper is the notion of the quasi-orthogonality applied to polynomials with complex coefficients; namely, we say that the given sequence of complex polynomials in a real argument is quasi-orthogonal if there is a complex-valued weight such that the integral of the product of two polynomials multiplied by this weight vanishes for the specially selected integration interval. Our discovery of the exceptional quasi-orthogonal sequences of polynomials with complex coefficients was stimulated by Chen and Srivastava’s [34] observation that the orthogonality relations for both classical Jacobi and R-Jacobi polynomials can be extended to the complex field. While the formal ‘complexification’ of the Jacobi indexes preserving the conventional orthogonality relations between the classical Jacobi polynomials [39] is a well-established (though relatively little-known) fact [40], the broadening of this assertion to include the finite orthogonal sequences [34] constitutes a significant achievement mostly overlooked in the literature.

2. The q-RSs of the JRef CSLE with the Simple-Pole Density Function

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

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

(1)

with the single pole density function

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

(2)

and the reference polynomial fraction (RefPF) parameterized as follows:

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

(3)

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

(4)

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

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

(5)

The sign of the spectral parameter ε is chosen to be positive (negative) when the Sturm–Liouville problem in question is formulated on the finite interval $-1<\mathsf{\eta }<1$ (or, respectively, on the positive infinite interval $1<\mathsf{\eta }<\infty$). An analysis of solutions of the CSLE (1) on the negative infinite interval $-\infty <\mathsf{\eta }<-1$ can be skipped without the loss of generality due to the symmetry of the RefPF (2) under reflection of its argument, accompanied by the interchange of the ExpDiffs ${\mathsf{\lambda }}_{\mathrm{o};\pm }$ for the CSLE poles at ±1.

As initially pointed to in [27], the JRef CSLE with the density function (2) has four infinite sequences of the q-RSs

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

(6)

at the energies

$$\mathsf{\epsilon }=sgn(1-{\mathsf{\eta }}^2){\mathsf{\epsilon }}_{\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}),$$

with

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

(7)

It can be verified that the shift parameters λ in Odake–Sasaki’s recipe [41] for constructing the rational SUSY partners of the t- and h-PT potentials are nothing but the indexes $\mathsf{\alpha }={\mathsf{\lambda }}_{+},$ $\mathsf{\beta }={\mathsf{\lambda }}_{-}$ 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 recently suggested [10] unified approach to the $\mathrm{R}\mathrm{D}\text{T}\mathrm{s}$ of these potentials using the first-degree (m = 1) seed polynomials. The shifts in the parameters λ under the rational Darboux–Crum transformations (RDCTs) in Odake–Sasaki’s approach are simply realizations of the changes in the ExpDiffs for the poles of the transformed RCSLEs under the action of sequential RRZTs after each RCSLE is converted to the Schrödinger equation by the corresponding Liouville transformation.

As clarified in [42], the equivalence relations derived in [43] for the $\mathrm{R}\mathrm{D}\mathrm{C}\text{T}\mathrm{s}$ of the translationally shape-invariant (TSI) potential of group A (in Odake–Sasaki’s [44] classification scheme of rational TSI potentials) constitute the intrinsic feature of the form-invariant RCSLEs, with the present analysis [11,12,18] focusing solely on the Fuchsian SLEs with the energy-independent ExpDiffs for the poles in the finite plane. The conversion of the JRef CSLE (1) to the Schrödinger equation before presenting the appropriate argumentation and then the return to the quasi-rational form of the solutions of the latter equation [43] seem to be an artificial aberration.

Note that Gómez-Ullate et al. [43] 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 [43] are allowed to take only positive values other than 1. (In the cases ${\mathsf{\lambda }}_{\mathrm{o};\pm }=1$, the corresponding second-order pole disappears, and the resultant problem with only the simple pole at ±1 requires special attention.) It will be shown in Section 6.1 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 {\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ (in contrast with the constraint imposed on the Jacobi indexes α and β in [43].

By choosing [39],

$${\mathsf{\lambda }}_{-},{\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-}+{\mathsf{\lambda }}_{+}+\mathrm{m}\ne -\mathrm{k}\mathrm{for}\mathrm{any}\mathrm{positive}\mathrm{integer}\mathrm{k}\le \mathrm{m}$$

(8)

(cf. (102) in [45] or (88) in [46]), we assure that the Jacobi polynomials in question have 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 thoroughly examined by us in [47]. 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, and, as a result, the ExpDiffs for the CSLE poles at ±1 become energy-independent [27]. Consequently, the polynomial components of the q-RSs (6) satisfy the eigenequation

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

(9)

with the dot standing for the derivative with respect to η. The vital point is that the linear coefficient function is independent of the polynomial degree, in the sharp contrast with the general case [47]. Note that the derivation of this equation was essentially based on the presumption that the quasi-rational function

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

(10)

satisfies the first-order differential equation:

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

(11)

In other words,

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

(12)

where the symbolic expression ld f[η] denotes the logarithmic derivative of the function f[η].

Re-writing the CSLE

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

(13)

in the Riccati form gives

$${\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]=-l\stackrel{•}{d}{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]+l{d}^2{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]+{\mathsf{\epsilon }}_0(\stackrel{\rightharpoonup }{\mathsf{\lambda }})\mathit{\rho }[\mathsf{\eta }]$$

(14)

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 ${\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$. In the following section, we will use a similar representation for the RefPFs of the rational Rudjak–Zakhariev transforms ($\mathrm{R}\mathrm{R}\mathrm{Z}\text{T}\mathrm{s}$) of the JRef CSLE (1) using the quasi-rational TFs (6). By exploiting the formalism put forward at the end of the last century by Rudjak and Zakhariev [24], we formulate the self-contained theory of the rational potentials exactly solvable by polynomials, which provides the rigorous mathematical basis for the traditional supersymmetric approach [5] utilized in [7,8,9,10].

3. Rational Rudjak–Zakhariev Transforms of JRef CSLE

Each of the q-RSs (6) can be used as the TF for the RRZT of the CSLE (1) defined via the requirement [26] that the quasi-rational function

$$*{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]={\displaystyle \frac{1}{\sqrt{\mathit{\rho }[\mathsf{\eta }]}{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}}=*{\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]/{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })$$

(15)

is the solution of the transformed CSLE

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

(16)

at the energy (7):

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

(17)

Representing the latter CSLE in the Riccati form:

$${\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]=-l\stackrel{•}{d}*{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]-l{d}^2*{\mathsf{\varphi }}_{\mathrm{m}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]-{\mathsf{\epsilon }}_{\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}){\mathit{\rho }}_{+}[\mathsf{\eta }]$$

(18)

similar to (14) and taking into account that the quasi-rational function

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

(19)

satisfies the Riccati equation (14) with m = 0 and ${\mathsf{\lambda }}_{\pm }$ increased by 1:

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

(20)

one finds

$$\begin{gathered} {\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|\mathrm{m}]={\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{\mathbf{1}}]+\stackrel{•}{ld}{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })-l{d}^2{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })+ \\ 2ld*{\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]ld{\mathrm{P}}_{\mathrm{m}}^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })-[{\mathsf{\epsilon }}_{\mathrm{m}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})-{\mathsf{\epsilon }}_0(-\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1})]{\mathit{\rho }}_{+}[\mathsf{\eta }]. \end{gathered}$$

(21)

For m = 1, the Jacobi polynomial in question takes the form

$${\mathrm{P}}_1^{({\mathsf{\lambda }}_{+},{\mathsf{\lambda }}_{-})}(\mathsf{\eta })\equiv {\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{-}+{\mathsf{\lambda }}_{+}+2)[\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})]$$

(22)

with

$${\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})={\displaystyle \frac{{\mathsf{\lambda }}_{-}-{\mathsf{\lambda }}_{+}}{{\mathsf{\lambda }}_{+}+{\mathsf{\lambda }}_{-}+2}}$$

(23)

And, therefore,

$$\stackrel{•}{ld}[\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})]=-{\displaystyle \frac{1}{{[\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}^2}}=-l{d}^2[\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})]$$

(24)

Taking into account that

$${\mathsf{\epsilon }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})-{\mathsf{\epsilon }}_0(-\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1})={\mathsf{\lambda }}_{-}+{\mathsf{\lambda }}_{+}+2$$

(25)

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

(26)

and decomposing the first-degree Jacobi polynomial on the right via (22), the RefPF of the RCSLE (17) can be represented for m = 1 as

$${\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|1]={\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };|\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{\mathbf{1}}|]-{\displaystyle \frac{2}{{[\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}^2}}+{\displaystyle \frac{2\mathsf{\eta }}{({\mathsf{\eta }}^2-1)[\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}}.$$

(27)

Examination of the RefPF (27) reveals that we deal with the canonical form of the Heun equation [4], and, therefore, we refer to the RCSLE (16) as the ‘restricted Heun-reference’ (restr-HRef), keeping in mind that the position of the third pole depends on the exponent parameters of the Heun equation.

Finally, let us draw the reader’s attention to the fact that the restr-HRef (27) is unambiguously determined by the indexes of the Jacobi polynomial representing the polynomial component of the quasi-rational TF (6), and, therefore, this must also be true for each of the Liouville potentials associated with the given SLP. This important observation by Lévai et al. [10] represents the interesting new element of the unified approach suggested by the cited authors for the restr-HRef potentials [8,9]. We shall come back to the discussion of this issue in Section 7.

4. Form-Invariance of Restr-HRef CSLE Under Two Sequential RZTs

Let us set apply the RRZT with the TF ${\mathsf{\varphi }}_1[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$ to the JRef CSLE (1). Setting

$$*{\mathsf{\lambda }}_{\mathrm{o};\pm }:=|{\mathsf{\lambda }}_{\pm }+1|$$

(28)

$${\mathsf{\sigma }}_{\pm }:=\mathit{sgn}({\mathsf{\lambda }}_{\pm }+1),$$

(29)

and substituting

$$\stackrel{\rightharpoonup }{\mathsf{\lambda }}=\stackrel{\rightharpoonup }{\mathsf{\sigma }}\times *{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}-\stackrel{\rightharpoonup }{1}$$

(30)

into (23) then gives

$${\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\sigma }}\times *{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}-\stackrel{\rightharpoonup }{1})={\displaystyle \frac{{\mathsf{\sigma }}_{-}*{\mathsf{\lambda }}_{\mathrm{o};-}-{\mathsf{\sigma }}_{+}*{\mathsf{\lambda }}_{\mathrm{o};+}}{{\mathsf{\sigma }}_{+}*{\mathsf{\lambda }}_{\mathrm{o};+}+{\mathsf{\sigma }}_{-}*{\mathsf{\lambda }}_{\mathrm{o};-}}}=$$

(31)

$${\mathrm{e}}_3^{\mathsf{\sigma }}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}):={\displaystyle \frac{\mathsf{\sigma }*{\mathsf{\lambda }}_{\mathrm{o};+}-*{\mathsf{\lambda }}_{\mathrm{o};-}}{\mathsf{\sigma }*{\mathsf{\lambda }}_{\mathrm{o};-}+*{\mathsf{\lambda }}_{\mathrm{o};+}}}$$

(32)

where

$$\mathsf{\sigma }=\mathit{sgn}({\displaystyle \frac{{\mathsf{\lambda }}_{-}+1}{{\mathsf{\lambda }}_{+}+1}}).$$

(33)

Making use of the conventional parameter $b(\mathsf{\beta },\mathsf{\alpha })$ defined via (A37) in Appendix C, we can alternatively represent the position (32) of the third pole of the RefPF (27) as

$${\mathrm{e}}_3^{\mathsf{\sigma }}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})=1/b^{\mathsf{\sigma }1}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})$$

(34)

By applying the RRZT with the TF ${\mathsf{\varphi }}_1[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$ to the JRef CSLE (1), we come to the two RCSLEs

$$\left\{{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{\mathsf{\eta }}^2}}+{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\text{*}{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|\mathsf{\sigma }]+|\mathsf{\epsilon }|\mathit{\rho }[\mathsf{\eta }]\right\}\mathsf{\Phi }[\mathsf{\eta };\text{*}{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\mathsf{\epsilon }|\mathsf{\sigma }]=0(\mathsf{\sigma }=\pm )$$

(35)

with the RefPFs

$${\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\text{*}{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|\mathsf{\sigma }]:={\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };\text{*}{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]-{\displaystyle \frac{2}{{[\mathsf{\eta }-{\mathrm{e}}_3^{\mathsf{\sigma }}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})]}^2}}+{\displaystyle \frac{2\mathsf{\eta }}{({\mathsf{\eta }}^2-1)[\mathsf{\eta }-{\mathrm{e}}_3^{\mathsf{\sigma }}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})]}}(\mathsf{\sigma }=\pm ).$$

(36)

It directly follows from (32) that

$$-1<{\mathrm{e}}_3^{+}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})<+1$$

(37)

and

$$|{\mathrm{e}}_3^{-}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})|>1,$$

(38)

respectively.

If the third pole lies outside the interval [−1,+1], we also choose

$$\text{*}{\mathsf{\lambda }}_{\mathrm{o};+}>\text{*}{\mathsf{\lambda }}_{\mathrm{o};-}$$

(39)

thereby assuring that it lies on the negative semi-axis:

$${\mathrm{e}}_3^{-}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})<-1.$$

(40)

By defining $\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ via (30), we find that the two sequential RRZTs with the TFs

$$\text{*}{\mathsf{\varphi }}_1[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]={\mathit{\rho }}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]/{\mathsf{\varphi }}_1[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$$

(41)

and ${\mathsf{\varphi }}_1[\mathsf{\eta };-\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$ convert the restr-HRef CSLE with the RefPF (27) into another restr-HRef CSLE with the RefPF

$${\mathrm{I}}_{\mathsf{\upsilon }}^{\mathrm{o}}[\mathsf{\eta };|1-{\mathsf{\lambda }}_{-}|,|1-{\mathsf{\lambda }}_{+}|]={\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };-\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{\mathbf{1}}]-{\displaystyle \frac{2}{{[\mathsf{\eta }-{\mathsf{\eta }}_1(-\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}^2}}+{\displaystyle \frac{2\mathsf{\eta }}{({\mathsf{\eta }}^2-1)[\mathsf{\eta }-{\mathsf{\eta }}_1(-\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}}.$$

(42)

It then directly follows from the analysis presented in Appendix A that this is also true for the second-order RDCT with the seed functions (41) and

$$\mathsf{\varphi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|1;--,1]:=\text{*}{\mathsf{\varphi }}_1[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]\mathrm{W}\{{\mathsf{\varphi }}_1[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}],{\mathsf{\varphi }}_1[\mathsf{\eta };-\stackrel{\rightharpoonup }{\mathsf{\lambda }}]\}.$$

(43)

We say that the restr-HRef CSLE (35) is double-step form-invariant iff the RDCT in question keeps the position of the third pole within the intervals (37) or (38), respectively. In other words, the exponents $-\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ of the TF ${\mathsf{\varphi }}_1[\mathsf{\eta };-\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$ must obey the constraint

$$\mathit{sgn}({\displaystyle \frac{1-{\mathsf{\lambda }}_{-}}{1-{\mathsf{\lambda }}_{+}}})=\mathsf{\sigma }.$$

(44)

Theorem 1. 

The restr-HRef CSLE is a double-step form-invariant provided that both ExpDiff $*{\mathsf{\lambda }}_{\mathrm{o};\pm }$ are larger than 2.

Proof. 

Representing (30) as

$$1-{\mathsf{\lambda }}_{-}=2-{\mathsf{\sigma }}_{-}*{\mathsf{\lambda }}_{\mathrm{o};-},1-{\mathsf{\lambda }}_{+}=2-{\mathsf{\sigma }}_{+}*{\mathsf{\lambda }}_{\mathrm{o};+},$$

(45)

let us start with the case ${\mathsf{\sigma }}_{-}=-,{\mathsf{\sigma }}_{+}=-\mathsf{\sigma }=\pm$

$$1-{\mathsf{\lambda }}_{-}=2+*{\mathsf{\lambda }}_{\mathrm{o};-},1-{\mathsf{\lambda }}_{+}=2+\mathsf{\sigma }*{\mathsf{\lambda }}_{\mathrm{o};+}.$$

(46)

One can directly verify that the parameters (46) are both positive if $\mathsf{\sigma }=+$ while having the opposite signs if $\mathsf{\sigma }=-$ and $*{\mathsf{\lambda }}_{\mathrm{o};+}>2$. □

Furthermore, examination of the parameters

$$1-{\mathsf{\lambda }}_{-}=2-*{\mathsf{\lambda }}_{\mathrm{o};-},1-{\mathsf{\lambda }}_{+}=2-\mathsf{\sigma }*{\mathsf{\lambda }}_{\mathrm{o};+}\mathrm{f}\mathrm{o}\mathrm{r}{\mathsf{\sigma }}_{-}=+,{\mathsf{\sigma }}_{+}=\mathsf{\sigma }=\pm$$

(47)

reveals that they have opposite signs if $*{\mathsf{\lambda }}_{\mathrm{o};-}>2$ and $\mathsf{\sigma }=-$, regardless of the value of the second ExpDiff $*{\mathsf{\lambda }}_{\mathrm{o};+}$. Both parameters are negative if $\mathsf{\sigma }=+$ and $*{\mathsf{\lambda }}_{\mathrm{o};\pm }>2$, which completes the proof. □

As seen from the presented proof, the restr-HRef CSLE (1) is also a double-step form-invariant for $*{\mathsf{\lambda }}_{\mathrm{o};\pm }<2$ if ${\mathsf{\sigma }}_{-}={\mathsf{\sigma }}_{+}=+$. There are three anomalous instances:

(i).

${\mathsf{\lambda }}_{-}=-*{\mathsf{\lambda }}_{\mathrm{o};-}-1,{\mathsf{\lambda }}_{+}=*{\mathsf{\lambda }}_{\mathrm{o};+}-1(\mathsf{\sigma }={\mathsf{\sigma }}_{-}=-,{\mathsf{\sigma }}_{+}=+,*{\mathsf{\lambda }}_{\mathrm{o};+}<2)$,

(ii).

${\mathsf{\lambda }}_{-}=*{\mathsf{\lambda }}_{\mathrm{o};-}-1,{\mathsf{\lambda }}_{+}=-*{\mathsf{\lambda }}_{\mathrm{o};+}-1(\mathsf{\sigma }=-,{\mathsf{\sigma }}_{\mp }=\pm ,*{\mathsf{\lambda }}_{\mathrm{o};-}<2)$,

(iii).

${\mathsf{\lambda }}_{\mp }=*{\mathsf{\lambda }}_{\mathrm{o};\mp }-1\mathrm{i}\mathrm{f}\mathsf{\sigma }={\mathsf{\sigma }}_{-}={\mathsf{\sigma }}_{+}=+,(*{\mathsf{\lambda }}_{\mathrm{o};+}-2)(*{\mathsf{\lambda }}_{\mathrm{o};-}-2)<0$,

when the second-order DCT in question violates the constraint (44). It can be shown in Section 6 that the restr-HRef CSLE (35) has a discrete energy spectrum if the quantization is performed on the intervals [−1,1] and [+1,∞) for σ = − and σ = +, respectively, provided that $*{\mathsf{\lambda }}_{\mathrm{o};-}>*{\mathsf{\lambda }}_{\mathrm{o};+}$ for σ, i.e., we can re-formulate the latter case as

$$(\mathrm{iii}).{\mathsf{\lambda }}_{\mp }=*{\mathsf{\lambda }}_{\mathrm{o};\mp }-1\mathrm{i}\mathrm{f}\mathsf{\sigma }={\mathsf{\sigma }}_{-}={\mathsf{\sigma }}_{+}=+,*{\mathsf{\lambda }}_{\mathrm{o};+}<2,*{\mathsf{\lambda }}_{\mathrm{o};-}>2.$$

One can easily verify that the ExpDiff ${\mathsf{\lambda }}_{\mathrm{o};+}=|*{\mathsf{\lambda }}_{\mathrm{o};+}-1|$ and ${\mathsf{\lambda }}_{\mathrm{o};-}=|*{\mathsf{\lambda }}_{\mathrm{o};-}-1|$ for the corresponding pole of the JRef CSLE in the finite plane lie within the LC range $(0<{\mathsf{\lambda }}_{\mathrm{o};\pm }<1)$ for the instance (i) and (ii), respectively. Examination of (46) and (47) with σ = − in both cases reveals that the second RDT with the TF ${\mathsf{\varphi }}_1[\mathsf{\eta };-\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$ converts the JRef CSLE into the restr-HRef CSLE with the ExpDiff

$$|1-{\mathsf{\lambda }}_{\mathrm{o};\pm }|=|*{\mathsf{\lambda }}_{\mathrm{o};\pm }-2|<1$$

for the pole at ±1, with ‘+’ or ‘−’ used for the instance (i) or accordingly (ii). In other words, this transformation keeps the given ExpDiff within the LC region and, as a result, violates the conventional rules of the SUSY quantum mechanics [28].

Similarly, the ExpDiff ${\mathsf{\lambda }}_{\mathrm{o};+}=|*{\mathsf{\lambda }}_{\mathrm{o};+}-1|$ pole of the JRef CSLE at +1 lies within the LC range $(0<{\mathsf{\lambda }}_{\mathrm{o};+}<1)$ for the instance (iii). Examination of (47) with σ = + reveals that the second RDT with the TF ${\mathsf{\varphi }}_1[\mathsf{\eta };-\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$ converts the JRef CSLE into the restr-HRef CSLE with the ExpDiff

$$|1-{\mathsf{\lambda }}_{\mathrm{o};+}|=|*{\mathsf{\lambda }}_{\mathrm{o};+}-2|<1$$

for the pole at +1 and therefore keeps the given ExpDiff within the LC region. Again, as discovered in [28], the RDTs of this kind violate the conventional rules of the SUSY quantum mechanics. We shall come back to this issue in Section 6.

5. Pseudo-Wronskian Representation of X₁-Jacobi DPSs

The main purpose of this section is to convert the transformed RCSLE (35) into the Bochner-type ODEs with polynomial solutions. It is shown that there are two distinguished polynomial sequences satisfying the constructed ODEs. Since these sequences either do not start from a constant or lack the first-degree polynomial, the polynomial solutions in question are not covered by the Bochner theorem [14], and we refer to these sequences as the X₁-Jacobi DPSs, by analogy with the term ‘X₁-Jacobi OPS’ brought in [15] for the infinite orthogonal subset of one of them. The most important element of the proofs presented below is that the X₁-Jacobi DPSs instigated in such a way are formed by the pseudo-Wronskians of two Jacobi polynomials introduced in the breakthrough analysis of the $\mathrm{D}\mathrm{C}\text{T}\mathrm{s}$ of the t-PT potential by Gómez-Ullate et al. [43].

5.1. RRZTs of JS Solutions

To proceed in the aforementioned direction, let us first demonstrate that the restr-HRef CSLEs (35) have infinite sets of the q-RSs obtained by acting the RRZT with the TF ${\mathsf{\varphi }}_1[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$ on various JS solutions. To be more precise, it will be confirmed that the $\mathrm{R}\mathrm{R}\mathrm{Z}\text{T}$ of the ${\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]$ with the eigenvalue ${\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime })$ has a quasi-rational form in the four quadrants ${\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}}^{\prime }$ of the vector parameter

$$(|{{\mathsf{\lambda }}^{\prime }}_{\pm }|=|{\mathsf{\lambda }}_{\pm }|>0)$$

(48)

such that the polynomial component of this solution coincides with the pseudo-Wronskian of the first- and j-th degree Jacobi polynomials with the indexes $\stackrel{\rightharpoonup }{\mathsf{\lambda }}$ and ${\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }$. Namely, the $\mathrm{R}\mathrm{R}\mathrm{Z}\text{T}$ of our interest is given by the generic formula [24]

$$\mathsf{\varphi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|1;\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}},\mathrm{j}]:=*{\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}]{\displaystyle \frac{\mathrm{W}\{{\mathsf{\varphi }}_1[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}],{\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]\}}{\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})}}$$

(49)

Theorem 2. 

The restr-HRef CSLE (35) has four infinite sequences of q-RSs with the polynomial components formed by pseudo-Wronskians of two Jacobi polynomials.

Proof. 

Representing the Wronskian in the right-hand side of (49) as

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

(50)

and also taking into account that

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

(51)

where, according to (91) in [43], with $\mathsf{\alpha }={\mathsf{\lambda }}_{+},\mathsf{\beta }={\mathsf{\lambda }}_{-}$, and n = 1,

$${\mathrm{d}}_{\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}},1}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})=\{\begin{array}{cc}{\mathsf{\lambda }}_{-}+1& \mathrm{i}\mathrm{f}\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=-+,\\ {\mathsf{\lambda }}_{+}+1& \mathrm{i}\mathrm{f}\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=+-,\\ 4& \mathrm{i}\mathrm{f}\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=--,\\ {\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{+}+{\mathsf{\lambda }}_{-}+2)& \mathrm{if}\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=++,\end{array}$$

(52)

we introduce the pseudo-Wronskian polynomials via the relations:

$$\begin{gathered} {\text{P}}_{1,\mathrm{j}+1-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}]:= \\ \left|\begin{array}{c}{\displaystyle \prod _{\aleph =\mp }(}\mathsf{\eta }-\aleph 1{)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}(1-{\cancel{\mathsf{\sigma }}}_{\aleph })}{\mathrm{P}}_1^{({\cancel{\mathsf{\sigma }}}_{+}{{\mathsf{\lambda }}^{\prime }}_{+},{\cancel{\mathsf{\sigma }}}_{-}{{\mathsf{\lambda }}^{\prime }}_{-})}(\mathsf{\eta }){\mathrm{P}}_{\mathrm{j}}^{({{\mathsf{\lambda }}^{\prime }}_{+},{{\mathsf{\lambda }}^{\prime }}_{-})}(\mathsf{\eta })\hfill \\ {\mathrm{d}}_{\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}},1}(\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}\times {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }){\mathrm{P}}_1^{({\cancel{\mathsf{\sigma }}}_{+}{{\mathsf{\lambda }}^{\prime }}_{+}+{\cancel{\mathsf{\sigma }}}_{+}1,{\cancel{\mathsf{\sigma }}}_{-}{{\mathsf{\lambda }}^{\prime }}_{-}+{\cancel{\mathsf{\sigma }}}_{-}1)}(\mathsf{\eta }){\stackrel{•}{\mathrm{P}}}_{\mathrm{j}}^{({{\mathsf{\lambda }}^{\prime }}_{+},{{\mathsf{\lambda }}^{\prime }}_{-})}(\mathsf{\eta })\hfill \end{array}\right| \end{gathered}$$

(53)

and then re-write the Wronskian (50) in the following quasi-rational form:

$$\begin{gathered} \mathrm{W}\{{\mathsf{\varphi }}_1[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}],{\mathsf{\varphi }}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]\}={\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\mathbf{1}-\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{\mathbf{1}}]{\mathsf{\varphi }}_0[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }]\times \\ {\text{P}}_{1,\mathrm{j}+1-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}]. \end{gathered}$$

(54)

Substituting (54) into solution (49) and making use of (41) thus gives the following:

$$\mathsf{\varphi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|1;\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}},\mathrm{j}]\propto {\displaystyle \frac{{\mathsf{\varphi }}_0[\mathsf{\eta };\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times (\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{1})]}{\mathsf{\eta }-{\mathsf{\eta }}_1(\stackrel{\rightharpoonup }{\mathsf{\lambda }})}}{\mathrm{P}}_{\mathrm{j}+1-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\mathsf{\lambda }}|1;\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{j}],$$

(55)

which completes the proof of Theorem 2. □

Comparing (54) with (A18) for $\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=--$ and m = 1, one finds that

$${\text{P}}_{1,\mathrm{j}+2}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|--]={\text{D}}_{\mathrm{j}+2}[\mathsf{\eta };-{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime },\mathbf{1}\mathbf{;}{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime },\mathrm{j}]$$

(56)

Substituting (A19) into the polynomials (A20) evaluated at $\mp 1$

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

(57)

gives

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

(58)

We thus proved that the PD (56) remains finite at $\mp 1$, assuming this is true for the j-th degree Jacobi polynomial on the right. Out of the four PD sequences introduced by us in [11] to generate the X_m-Jacobi DPSs, it was the only sequence formed by the polynomials remaining finite at ±1 and therefore spanning the DPS referred to by us for this reason as the X_m-Jacobi DPS of series D.

Representing the pseudo-Wronskians (53) for $\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=\mp \pm$ as

$$\begin{gathered} {\text{P}}_{1,\mathrm{j}+1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|\mp \pm ]=(\mathsf{\eta }\pm 1){\mathrm{P}}_1^{(\pm {{\mathsf{\lambda }}^{\prime }}_{+},\mp {{\mathsf{\lambda }}^{\prime }}_{-})}(\mathsf{\eta }){\stackrel{•}{\mathrm{P}}}_{\mathrm{j}}^{({{\mathsf{\lambda }}^{\prime }}_{+},{{\mathsf{\lambda }}^{\prime }}_{-})}(\mathsf{\eta }) \\ +({{\mathsf{\lambda }}^{\prime }}_{\mp }-1){\mathrm{P}}_1^{(\pm {{\mathsf{\lambda }}^{\prime }}_{+}\pm 1,\mp {{\mathsf{\lambda }}^{\prime }}_{-}\mp 1)}(\mathsf{\eta }){\mathrm{P}}_{\mathrm{j}}^{({{\mathsf{\lambda }}^{\prime }}_{+},{{\mathsf{\lambda }}^{\prime }}_{-})}(\mathsf{\eta }) \end{gathered}$$

(59)

we come to the polynomials (A25) in Appendix B, with m set to 1. As demonstrated in Appendix C, the sequences (59) simply represent two alternative forms of the X₁-Jacobi DPS of series J.

For $\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=++$, the pseudo-Wronskian (53) turns into the polynomial Wronskian

$${\text{P}}_{1,\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|++]={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({{\mathsf{\lambda }}^{\prime }}_{+}+{{\mathsf{\lambda }}^{\prime }}_{-}+2)\mathrm{W}\{\mathsf{\eta }-{\mathsf{\eta }}_1({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }),{\mathrm{P}}_{\mathrm{j}}^{({{\mathsf{\lambda }}^{\prime }}_{+},{{\mathsf{\lambda }}^{\prime }}_{-})}(\mathsf{\eta })\}$$

(60)

Since each Wronskian in the sequence can be converted into the PD via (A58) in Appendix D, this polynomial sequence simply represents the alternative form of the aforementioned X₁-Jacobi DPS of series D.

5.2. X₁-Jacobi DPSs Composed of Pseudo-Wronskians of Two Jacobi Polynomials

Let us now draw the reader’s attention to the fact that the absolute values of the vector parameters

$$*\stackrel{\rightharpoonup }{\mathsf{\lambda }}:=\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times (\stackrel{\rightharpoonup }{\mathsf{\lambda }}+\stackrel{\rightharpoonup }{1})\equiv {\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }+\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{1}$$

(61)

specifying the power exponents of the q-RS (55) represent the ExpDiffs

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

for the poles of the RCSLE (35) at ±1, and, therefore, each pseudo-Wronskian polynomial remains finite at both poles (at least as far as the indexes of the j-th degree Jacobi polynomial in the right-hand side of (58) do not violate the constraints (8), with m and ${\mathsf{\lambda }}_{\pm }$ replaced for j and ${\mathsf{\lambda }}_{\pm }^{\prime }$, respectively).

Setting

$$*\mathsf{\sigma }:=\mathit{sgn}(*{\mathsf{\lambda }}_{+}*{\mathsf{\lambda }}_{-})$$

(62)

and taking into account (32), one finds

$${\mathrm{e}}_3^{*\mathsf{\sigma }}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})=1/b(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}).$$

(63)

At this point, it is convenient to explicitly re-write the CSLEs (35) as the canonical forms of the Heun equations

$$\left\{{\displaystyle \frac{d^2}{d{\mathsf{\eta }}^2}}+{\mathrm{I}}_{\mathrm{H}}[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]+|\mathsf{\epsilon }|\mathit{\rho }[\mathsf{\eta }]\right\}\mathsf{\Phi }[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathsf{\epsilon }]=0,$$

(64)

where we set

$${\mathrm{I}}_{\mathrm{H}}[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]:={\mathrm{I}}_{*\mathsf{\sigma }}^{\mathrm{o}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]$$

(65)

$$={\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]-{\displaystyle \frac{2}{{[\mathsf{\eta }-b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}^2}}+{\displaystyle \frac{2\mathsf{\eta }}{({\mathsf{\eta }}^2-1)[\mathsf{\eta }-b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}}$$

(66)

Note that

$$b(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})=b(-*\stackrel{\rightharpoonup }{\mathsf{\lambda }})$$

(67)

and, therefore, the restr-Heun equation (64) is invariant under the simultaneous change in the signs of both parameters $*{\mathsf{\lambda }}_{\pm }$:

$${\mathrm{I}}_{\mathrm{H}}[\mathsf{\eta };-*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]={\mathrm{I}}_{\mathrm{H}}[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]$$

(68)

Since the polynomial sequences (58) and (59) start from the first-degree polynomial while the polynomial sequence (60) lacks a first-degree polynomial, all the mentioned polynomial sequences violate the prerequisites of the Bochner theorem. We thus arrived to one of the most important results of this paper.

Theorem 3. 

The polynomial sequences (58), (59) and (60) satisfy the Bochner-type ODEs with polynomial coefficients and, therefore, form the X-DPSs, which either start from a first-degree polynomial or lack a polynomial of this degree.

Proof. 

First note that the function (55),

$$\begin{gathered} \mathsf{\varphi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times *\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}|1;\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}},\mathrm{j}]\propto {\displaystyle \frac{{\mathsf{\varphi }}_0[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}{\mathsf{\eta }-b_{\mathbf{*}\mathsf{\sigma }}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})}}\times \\ {\text{P}}_{\mathrm{j}+1-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times *\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}|1;\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{j}], \end{gathered}$$

(69)

is the solution of the ODE

$$\left\{{\displaystyle \frac{d^2}{d{\mathsf{\eta }}^2}}+{\mathrm{I}}_{\mathrm{H}}[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]+{\mathsf{\epsilon }}_{\mathrm{j}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{1})\mathit{\rho }[\mathsf{\eta }]\right\}\mathsf{\varphi }[\mathsf{\eta };\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times *\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}|1;\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}},\mathrm{j}]=0,$$

(70)

where the vector parameter ${\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }$, specifying the corresponding eigenvalue, was replaced for $*\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{1}$, based on the definition (61) of $\stackrel{\rightharpoonup }{\mathsf{\lambda }}*$. Taking into account that, according to (27),

$$\begin{gathered} {\displaystyle \frac{\mathsf{\eta }-b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})}{{\mathsf{\varphi }}_0[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}}{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{\mathsf{\eta }}^2}}{\displaystyle \frac{{\mathsf{\varphi }}_0[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}{\mathsf{\eta }-b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})}}=-{\mathrm{I}}_{\mathrm{H}}[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]- \\ {\displaystyle \frac{2{\mathrm{P}}_1^{(*{\mathsf{\lambda }}_{+}-1,*{\mathsf{\lambda }}_{-}-1)}(\mathsf{\eta })}{({\mathsf{\eta }}^2-1)[\mathsf{\eta }-b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}}+{\displaystyle \frac{{\mathsf{\epsilon }}_0(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})}{{\mathsf{\eta }}^2-1}} \end{gathered}$$

(71)

we come to the Bochner-type ODEs with the polynomial coefficients:

$$\begin{gathered} \{\mathbf{D}[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})]+{\mathrm{C}}_1[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathsf{\Delta }{\mathsf{\epsilon }}_{\mathrm{j}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}})]\}\times \\ {\text{P}}_{\mathrm{j}+1-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times *\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}|1;\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{j}]=0, \end{gathered}$$

(72)

where the coefficient function of the first derivative in the second-order differential expression

$$\mathbf{D}[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathrm{r}]:=({\mathsf{\eta }}^2-1)[\mathsf{\eta }-\mathrm{r}]{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{\mathsf{\eta }}^2}}+2{\mathrm{B}}_2[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathrm{r}]{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}$$

(73)

is represented by the second-degree polynomial

$${\mathrm{B}}_2[\mathsf{\eta };\mathsf{\beta },\mathsf{\alpha };\mathrm{r}]:=(\mathsf{\eta }-\mathrm{r}){\mathrm{P}}_1^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta })+1-{\mathsf{\eta }}^2$$

(74)

As for the free term of the ODE (72), it portrays a form of the first-degree polynomial linear in the energy

$${\mathrm{C}}_1[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\mathrm{r};\mathsf{\epsilon }]:=-2{\mathrm{P}}_1^{(*{\mathsf{\lambda }}_{+}-1,*{\mathsf{\lambda }}_{-}-1)}(\mathsf{\eta })-\mathsf{\epsilon }(\mathsf{\eta }-\mathrm{r})$$

(75)

and ε equal to

$$\mathsf{\Delta }{\mathsf{\epsilon }}_{\mathrm{j}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}):={\mathsf{\epsilon }}_{\mathrm{j}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}\times \stackrel{\rightharpoonup }{\mathbf{1}})-{\mathsf{\epsilon }}_0(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})$$

(76)

The important common feature of the ODEs (72) is that the leading coefficient polynomial of degree three has only simple zeros. This is the direct consequence of the fact that each exponent of the quasi-rational function (69) coincides with one of the ChExps for the corresponding pole. □

Recently, Masjed-Jamei et al. [48,49] made an attempt to extend the family of the Pearson distributions to the theory of the X₁-OPSs. To relate the Bochner-type equations (72) to the ODE (37) in [49], let us first remind the reader that the relevant parts of Hildebrandt’s analysis [50] of the classical DPSs associated with the Pearson differential equation (including the celebrated Pearson differential equation itself) were, to a large extent, a rediscovery of Routh’ results [51], which were shamefully overlooked in the 20th century mathematical literature, with Al-Salam’s paper [52] as the only exception known to us.

Let us define the conventional Routh function via the relation [51,53]:

$${\mathrm{R}}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]:=\mathsf{\sigma }[\mathsf{\eta }]{\mathsf{\omega }}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }],$$

(77)

with

$$\mathsf{\sigma }[\mathsf{\eta }]:=1-{\mathsf{\eta }}^2$$

and

$${\mathsf{\omega }}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]={(1-\mathsf{\eta })}^{\mathsf{\alpha }}{(1+\mathsf{\eta })}^{\mathsf{\beta }}\mathrm{for}-1<\mathsf{\eta }<+1$$

(78)

Masjed-Jamei et al. [48,49] suggested to generalize (77) as follows:

$${\stackrel{⌢}{\mathrm{R}}}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]:=\mathsf{\sigma }[\mathsf{\eta }]{\stackrel{⌢}{\mathsf{\omega }}}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }],$$

(79)

where

$${\stackrel{⌢}{\mathsf{\omega }}}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]:={\displaystyle \frac{{\mathsf{\omega }}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]}{{(\mathsf{\eta }-\mathrm{r})}^2}},$$

(80)

with r standing for the extraneous pole of the Heun equation. Taking into account that

$${\stackrel{⌢}{\mathrm{R}}}_{*{\mathsf{\lambda }}_{-},*{\mathsf{\lambda }}_{-}}[\mathsf{\eta }]:={\displaystyle \frac{{\mathsf{\varphi }}_0^2[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}{{(\mathsf{\eta }-\mathrm{r})}^2}}$$

(81)

we find that the coefficient function of the first derivative in (73) and the generalized Routh function (80) are related in exactly the same way as the corresponding coefficient function ${\mathsf{\tau }}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]$ of the Jacobi equation and the conventional Routh function (77). Namely,

$$2{\mathrm{B}}_2[\mathsf{\eta };\mathsf{\beta },\mathsf{\alpha };\mathrm{r}]=-{\stackrel{⌢}{\mathsf{\tau }}}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]$$

(82)

with

$${\stackrel{⌢}{\mathsf{\tau }}}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]:={\displaystyle \frac{\mathsf{\sigma }[\mathsf{\eta }]}{{\stackrel{⌢}{\mathrm{R}}}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]}}{\displaystyle \frac{\mathrm{d}{\stackrel{⌢}{\mathrm{R}}}_{\mathsf{\alpha },\mathsf{\beta }}}{\mathrm{d}\mathsf{\eta }}}.$$

(83)

Though the second-degree polynomial (83) precisely matches the coefficient function of the ODE (37) in [49] for $\mathsf{\theta }=-2$, the first-degree polynomial (75) drastically differs from the free term of the latter equation. Indeed, representing the first-degree Jacobi polynomial in (75) as

$$2{\mathrm{P}}_1^{(\mathsf{\alpha }-1,\mathsf{\beta }-1)}(\mathsf{\eta })=(\mathsf{\alpha }+\mathsf{\beta })(\mathsf{\eta }-\mathrm{r})+(\mathsf{\alpha }+\mathsf{\beta })\mathrm{r}-\mathsf{\beta }+\mathsf{\alpha }$$

and taking into account that

$$\mathsf{\Delta }{\mathsf{\epsilon }}_{\mathrm{n}-1}(\mathsf{\beta },\mathsf{\alpha };\pm ,\mp )=(\mathrm{n}-1)(\mathsf{\alpha }+\mathsf{\beta }+\mathrm{n})$$

(84)

we find that the leading coefficient of the polynomial

$$\begin{gathered} -{\mathrm{C}}_1[\mathsf{\eta };\mathsf{\beta },\mathsf{\alpha };\mathrm{r};\mathsf{\Delta }{\mathsf{\epsilon }}_{\mathrm{n}-1}(\mathsf{\beta },\mathsf{\alpha };\pm \mp )]=[\mathrm{n}(\mathsf{\alpha }+\mathsf{\beta }+\mathrm{n}-1)-\mathrm{n}+1](\mathsf{\eta }-\mathrm{r}) \\ +(\mathsf{\alpha }+\mathsf{\beta })\mathrm{r}-\mathsf{\beta }+\mathsf{\alpha } \end{gathered}$$

(85)

agrees with that in the ODE (37) in [49] only for n = 1, while the free term of this polynomial is simply equal to

$$c_0^{(\mathrm{P})}=(\mathsf{\alpha }+\mathsf{\beta })\mathrm{r}-\mathsf{\beta }+\mathsf{\alpha }$$

in contrast with the cumbersome algebraic expression listed in [49]. And what is even more important, the cited equation may have polynomial solutions only if $\mathrm{r}=b(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})$—the issue simply ignored in [49]. Below, we just set

$${\mathrm{B}}_2[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};*\mathsf{\sigma }]:={\mathrm{B}}_2[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})]$$

(86)

$${\mathrm{C}}_1[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};*\mathsf{\sigma };\mathsf{\epsilon }]:={\mathrm{C}}_1[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }});\mathsf{\epsilon }]$$

(87)

getting rid of the argument r.

Let us explicitly confirm that the formulas derived by us for the polynomials (86) and (87) match the conventional formulas derived in [15] for the Bochner-type differential equations solved in terms of the X₁-Jacobi orthogonal polynomials. Introducing the parameters $a=(\mathsf{\beta }-\mathsf{\alpha })/2$, b, and c = b +1/a via (5a) and (5b) in [15] with

$$b\equiv b(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}),\mathrm{n}=\mathrm{j}+1,\mathsf{\alpha }\equiv *{\mathsf{\lambda }}_{+},\mathsf{\beta }\equiv *{\mathsf{\lambda }}_{-},*\mathsf{\sigma }=+,\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}}=\pm \mp$$

(88)

here, one finds

$${\mathrm{P}}_1^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta })=a(\mathrm{c}\mathsf{\eta }-1)=a(b\mathsf{\eta }-1)+\mathsf{\eta }$$

(89)

and

$${\mathrm{P}}_1^{(\mathsf{\alpha }-1,\mathsf{\beta }-1)}(\mathsf{\eta })=a(b\mathsf{\eta }-1)\equiv (\mathsf{\alpha }+\mathsf{\beta })[\mathsf{\eta }-b^{-1}(\mathsf{\beta },\mathsf{\alpha })]$$

(90)

The corresponding polynomial (74) takes the form

$${\mathrm{B}}_2[\mathsf{\eta };\mathsf{\beta },\mathsf{\alpha };+]={\stackrel{⌢}{\mathrm{P}}}_1^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta }){\mathrm{C}}_1[\mathsf{\eta };\mathsf{\beta },\mathsf{\alpha };+;0]$$

(91)

with

$${\stackrel{⌢}{\mathrm{P}}}_1^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta })\equiv -{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}(\mathsf{\eta }-\mathrm{c})$$

(92)

and

$${\mathrm{C}}_1[\mathsf{\eta };\mathsf{\beta },\mathsf{\alpha };+;0]=(\mathsf{\beta }-\mathsf{\alpha })[1-b(\mathsf{\beta },\mathsf{\alpha })\mathsf{\eta }]$$

(93)

(cf. (2.1) in [54]). Note that the energy shift (84) vanishes for the first-degree X₁-Jacobi polynomial (92), as expected. □

We have confirmed that our expression for the free term of the ODE (72) is correct, and the aforementioned discrepancy between this expression and the free term of the ODE (37) in [49] is simply the question of the deceptive terminology, i.e., the polynomials constructed in [48,49] using the generating function (80) are not the X₁-Jacobi polynomials in the conventional sense.

5.3. Bochner-Type ODEs in Heun Form

Our next step is to explicitly exploit the fact that the Bochner-type ODEs (72) represent the Fuchsian equation with the three regular singular points in the finite plane. As a result, the elementary change in variable $\mathsf{\eta }=2\mathrm{z}-1$ converts these equations to the Heun form [4]:

$$\begin{gathered} \{{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{\mathrm{z}}^2}}+{\displaystyle \frac{{\mathrm{B}}_2[\mathrm{z};*a(*\stackrel{\rightharpoonup }{\mathsf{\lambda }});*\stackrel{\rightharpoonup }{\mathsf{\lambda }};*\mathsf{\sigma }]}{\mathrm{z}(\mathrm{z}-1)[\mathrm{z}-*a(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{z}}}+{\displaystyle \frac{{\mathsf{\alpha }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}){\mathsf{\beta }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})\mathrm{z}-{\mathrm{q}}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}})}{\mathrm{z}(\mathrm{z}-1)[\mathrm{z}-*\mathrm{a}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})]}}\}\times \\ {\text{P}}_{\mathrm{n}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times *\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}|1;\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{n}-1+{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}]=0, \end{gathered}$$

(94)

where

$$*a(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}):={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}[b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})+1],$$

(95)

$${\mathrm{B}}_2[\mathrm{z};*a;*\stackrel{\rightharpoonup }{\mathsf{\lambda }};*\mathsf{\sigma }]:=2(\mathrm{z}-*a){\mathrm{P}}_1^{(*{\mathsf{\lambda }}_{+},*{\mathsf{\lambda }}_{-})}(2\mathrm{z}-1)-4\mathrm{z}(\mathrm{z}-1),$$

(96)

and

$${\mathsf{\alpha }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}){\mathsf{\beta }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})=-\mathrm{n}(*{\mathsf{\lambda }}_{-}+*{\mathsf{\lambda }}_{+}+\mathrm{n}-1)$$

(97)

Combining the latter formula with

$${\mathsf{\alpha }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})+{\mathsf{\beta }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})=*{\mathsf{\lambda }}_{+}+*{\mathsf{\lambda }}_{-}-1$$

(98)

gives

$${\mathsf{\alpha }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})\equiv -\mathrm{n},{\mathsf{\beta }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})=*{\mathsf{\lambda }}_{+}+*{\mathsf{\lambda }}_{-}+\mathrm{n}-1$$

(99)

Evaluating the leading coefficient of the second-degree polynomial (96) confirms that

$${\mathsf{\alpha }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}){\mathsf{\beta }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})=-\mathrm{n}[{\mathrm{B}}_{2;2}(*a(*\stackrel{\rightharpoonup }{\mathsf{\lambda }});*\stackrel{\rightharpoonup }{\mathsf{\lambda }};*\mathsf{\sigma })+\mathrm{n}-1]$$

(100)

in agreement with the general properties (1.1) and (1.2) of the Heine polynomials in [55]. The accessory parameter for the Heun equation (94) is given by the elementary formula

$${\mathrm{q}}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}})=-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathrm{C}}_1[-1;*\stackrel{\rightharpoonup }{\mathsf{\lambda }};*\mathsf{\sigma };\mathsf{\Delta }{\mathsf{\epsilon }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}})]$$

(101)

Coming back to the particular case (88) and evaluating the polynomial (93) at η = −1, one finds

$${\mathrm{C}}_1[-1;*\stackrel{\rightharpoonup }{\mathsf{\lambda }};+;0]=2*{\mathsf{\lambda }}_{-}$$

(102)

and therefore

$${\mathrm{C}}_1[-1;*\stackrel{\rightharpoonup }{\mathsf{\lambda }};+;\mathsf{\epsilon }]=2[*{\mathsf{\lambda }}_{-}+*a(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})\mathsf{\epsilon }]$$

(103)

Taking into account that the energy shift (84) vanishes for n = 1, we can re-write (101) as

$${\mathrm{q}}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\pm ,\mp )={\mathrm{q}}_1(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})-[(\mathrm{n}-1)(*{\mathsf{\lambda }}_{-}+*{\mathsf{\lambda }}_{+}+\mathrm{n})*a(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})]$$

(104)

with

$${\mathrm{q}}_1(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})=-*{\mathsf{\lambda }}_{-}.$$

(105)

Expressing the polynomial (93) in terms of z:

$${\mathrm{C}}_1[2\mathrm{z}-1;*\stackrel{\rightharpoonup }{\mathsf{\lambda }};+;0]=-2(*{\mathsf{\lambda }}_{-}-\mathbf{*}{\mathsf{\lambda }}_{+})[b(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})\mathrm{z}-*a(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})]$$

(106)

with

$$*a(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})={\displaystyle \frac{*{\mathsf{\lambda }}_{-}}{*{\mathsf{\lambda }}_{-}-*{\mathsf{\lambda }}_{+}}}$$

(107)

one can verify that

$${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathrm{C}}_1[2\mathrm{z}-1;*\stackrel{\rightharpoonup }{\mathsf{\lambda }};+;0]={\mathsf{\alpha }}_1(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}){\mathsf{\beta }}_1(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})\mathrm{z}-{\mathrm{q}}_1(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})$$

(108)

as expected.

Comparing (99) with the parameters α and β defined via (6.5) in [56], we assert that the latter formulas specify the Heun equation for the polynomials of degree k + 1 (not k as stated in [56]), with k standing for the integer j here. Re-writing (104) as

$${\mathrm{q}}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\pm ,\mp )=*a(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})[*{\mathsf{\lambda }}_{+}-*{\mathsf{\lambda }}_{-}-(\mathrm{n}-1)(*{\mathsf{\lambda }}_{-}+*{\mathsf{\lambda }}_{+}+\mathrm{n})]$$

(109)

and making use of (97), coupled with (98), we can represent the last term in the brackets in the right-hand side of (109) as

$$-(\mathrm{n}-1)({\mathsf{\lambda }}_{-}+{\mathsf{\lambda }}_{+}+\mathrm{n})={\mathsf{\alpha }}_{\mathrm{n}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }}){\mathsf{\beta }}_{\mathrm{n}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})+{\mathsf{\alpha }}_{\mathrm{n}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})+{\mathsf{\beta }}_{\mathrm{n}}(\stackrel{\rightharpoonup }{\mathsf{\lambda }})+1$$

(110)

which gives

$$\begin{gathered} {\mathrm{q}}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\pm ,\mp )=-{\displaystyle \frac{*{\mathsf{\lambda }}_{-}}{*{\mathsf{\lambda }}_{+}-*{\mathsf{\lambda }}_{-}}}\times \\ [*{\mathsf{\lambda }}_{+}-*{\mathsf{\lambda }}_{-}+{\mathsf{\alpha }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}){\mathsf{\beta }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})+{\mathsf{\alpha }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})+{\mathsf{\beta }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})+1]. \end{gathered}$$

(111)

By setting

$$\mathsf{\gamma }=*{\mathsf{\lambda }}_{-}+1,*{\mathsf{\lambda }}_{+}-*{\mathsf{\lambda }}_{-}={\mathsf{\alpha }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})+{\mathsf{\beta }}_{\mathrm{n}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})-2\mathsf{\gamma }+3$$

(112)

in (111), we come to the formula listed for the accessory parameter in [56].

Alternatively, one can re-write the Bochner-type ODE (72) as the eigenvalue problem

$$[{\mathbf{T}}_{*{\mathsf{\lambda }}_{+},*{\mathsf{\lambda }}_{-};1}-\mathsf{\Delta }{\mathsf{\epsilon }}_{\mathrm{j}}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }};\cancel{\stackrel{\rightharpoonup }{\mathsf{\sigma }}})]{\text{P}}_{\mathrm{j}+1-{\cancel{\mathsf{\sigma }}}_{+}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\cancel{\mathsf{\sigma }}}_{-}{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta };\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}\times *\stackrel{\rightharpoonup }{\mathsf{\lambda }}-\stackrel{\rightharpoonup }{1}|1;\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}},\mathrm{j}]=0$$

(113)

for the exceptional differential operator:

$${\mathbf{T}}_{*{\mathsf{\lambda }}_{+},*{\mathsf{\lambda }}_{-};1}(\mathsf{\eta }):=({\mathsf{\eta }}^2-1){\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{\mathsf{\eta }}^2}}-{\displaystyle \frac{{\mathrm{B}}_2[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }};*\mathsf{\sigma }]}{b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})-\mathsf{\eta }}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}+{\displaystyle \frac{2{\mathrm{P}}_1^{(*{\mathsf{\lambda }}_{+}-1,*{\mathsf{\lambda }}_{-}-1)}(\mathsf{\eta })}{b^{*\mathsf{\sigma }1}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})-\mathsf{\eta }}}$$

(114)

with the eigenvalues (76). We thus come to the renowned eigenequation [15,54] for the X₁-Jacobi polynomials (A36):

$$[{\mathbf{T}}_{\mathsf{\alpha },\mathsf{\beta };1}-4(\mathrm{n}-1)(\mathsf{\alpha }+\mathsf{\beta }+\mathrm{n})]{\stackrel{⌢}{\mathrm{P}}}_{\mathrm{n}}^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta })=0$$

(115)

though without any constraints on the relative sign of the parameters α and β. It should be nevertheless stressed that we list these operators only for the proper references and by no means need them in our analysis.

6. Energy Spectrum of ‘Prime’ SLEs Solved Under Dirichlet Boundary Conditions

We are finally ready to formulate the Sturm–Liouville problems for the restr-HRef CSLEs (35) with the boundary conditions imposed at the ends of either the finite interval [−1,+1] or the positive infinite interval [1,∞), assuming that the CSLE in question does not have poles inside the given interval. 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 necessarily pinpoints the PFS. As pointed to in [28] and illuminated below in the context of the restr-HRef CSLEs (35), the p-SLEs solved under the DBCs provide a useful tool to study the discrete energy spectra of the $\mathrm{R}\mathrm{R}\mathrm{Z}\text{T}\mathrm{s}$ of exactly solvable RCSLEs. While our conclusions concerning the infinite orthogonal reduction of the X₁-Jacobi DPS mostly duplicate the renowned results of the exhaustive theory of the X₁-Jacobi OPS [15,45,54], the extension of our formalism to the infinite interval [1,∞) represents the new remarkable development mostly ignored in the mathematical literature.

6.1. X₁-Jacobi OPS

Let us start from the generic SLE

$$\{-{\displaystyle \frac{d}{d\mathsf{\eta }}}\text{p}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]{\displaystyle \frac{d}{d\mathsf{\eta }}}+{\text{q}}^{\text{p}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]-\mathsf{\epsilon }{w}^{\text{p}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\}\cancel{\mathsf{\Psi }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\mathsf{\epsilon }]=0$$

(116)

obtained from the JRef CSLE (1) by the gauge transformation

$$\mathsf{\Phi }[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\mathsf{\epsilon }]={\text{p}}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]\cancel{\mathsf{\Psi }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\mathsf{\epsilon }]$$

(117)

Here, the PF representing the zero-energy free term has the following generic form [28]:

$${\text{q}}^{\text{p}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=\text{p}[\mathsf{\eta }]{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]+\text{I}\{\text{p}[\mathsf{\eta }])\},$$

(118)

with

$$\tilde{\text{I}}\{\text{p}[\mathsf{\eta }]\}:={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\stackrel{•}{\text{p}}}^2[\mathsf{\eta }]/\text{p}[\mathsf{\eta }]-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\stackrel{••}{\text{p}}[\mathsf{\eta }]$$

(119)

and the weight is related to the density function (2) via the generic formula

$${\mathrm{w}}^{\text{p}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=\text{p}[\mathsf{\eta }]\mathit{\rho }[\mathsf{\eta }].$$

(120)

The p-SLE

$$\{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}(1-{\mathsf{\eta }}^2){\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}-\cancel{\text{q}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]+\mathsf{\epsilon }\}\cancel{\mathsf{\Psi }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\mathsf{\epsilon }]=0$$

(121)

for the finite interval [−1,+1] is selected by choosing

$$\text{p}[\mathsf{\eta }]=1-{\mathsf{\eta }}^2\mathrm{f}\mathrm{o}\mathrm{r} \mathsf{\eta }\in (-1,+1),$$

(122)

with the zero-energy free term taking form [57]:

$$\cancel{\text{q}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=(1-{\mathsf{\eta }}^2){\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]+{(1-{\mathsf{\eta }}^2)}^{-1}.$$

(123)

Keeping in mind that

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

(124)

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};\pm }$ but differ by their sign. The PFS can be thus unambiguously determined by solving the p-SLE (121) under the DBCs

$$\underset{\mathsf{\eta }\to \pm 1}{lim}\cancel{\mathsf{\Psi }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};{\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})]=0$$

(125)

The solution of the formulated Dirichlet problem with an eigenvalue ${\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})$ can be represented in the form:

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

where the function ${\mathrm{F}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]$ is finite at the both endpoints:

$$0<\underset{\mathsf{\eta }\to \pm 1}{lim}|{\mathrm{F}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]|<\infty$$

satisfies the Jacobi differential equation

$$\begin{gathered} [({\mathsf{\eta }}^2-1){\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{\mathsf{\eta }}^2}}+2{\mathrm{P}}_1^{({\mathsf{\lambda }}_{\mathrm{o};+},{\mathsf{\lambda }}_{\mathrm{o}-})}(\mathsf{\eta }){\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}[{\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})-{\mathsf{\epsilon }}_0({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})]]\times \\ {\mathrm{F}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=0 \end{gathered}$$

(126)

This confirms the existence of the quasi-rational eigenfunctions of the form:

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

(127)

which are normalizable:

$${\displaystyle \underset{-1}{\overset{+1}{\int }}\mathrm{d}\mathsf{\eta }}{\cancel{\mathsf{\psi }}}_{\mathrm{j}}^2[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]<\infty$$

(128)

and mutually orthogonal [29]:

$${\displaystyle \underset{-1}{\overset{+1}{\int }}\mathrm{d}\mathsf{\eta }}{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]{\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=0(\mathrm{j}\ne {\mathrm{j}}^{\prime }).$$

(129)

The latter observation brings us to the conventional orthogonality relations for the classical Jacobi polynomials [39]

$$\begin{gathered} {\displaystyle \underset{-1}{\overset{+1}{\int }}\mathrm{d}\mathsf{\eta }}{(1+\mathsf{\eta })}^{{\mathsf{\lambda }}_{\mathrm{o};-}}{(1-\mathsf{\eta })}^{{\mathsf{\lambda }}_{\mathrm{o};+}}{\mathrm{P}}_{\mathrm{j}}^{({\mathsf{\lambda }}_{\mathrm{o};+},{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta }){\mathrm{P}}_{{\mathrm{j}}^{\prime }}^{({\mathsf{\lambda }}_{\mathrm{o};+},{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta })=0 \\ (\mathrm{j}\ne {\mathrm{j}}^{\prime }). \end{gathered}$$

(130)

Since the sequence of the eigenfunctions (127) starts from the nodeless function, each polynomial of degree j has exactly j zeros, and the corresponding eigenvalues are unrestricted from above, we assert that the Dirichlet problem in question does not have any solutions other than (127). Since the PFS of the Jacobi equation (126) remains finite at the given singular endpoint ±1, and the other Frobenius solution diverges as ${(1\mp \mathsf{\eta })}^{-{\mathsf{\lambda }}_{\mathrm{o};\pm }}$ as $\mathsf{\eta }\to \pm 1,$ we confirm that the DBCs (125) select all possible solutions of the Jacobi equation, which are both square-integrable with the weight (78) and finite at ±1 (cf. §9 to in [58]).

For ${\mathsf{\lambda }}_{\mathrm{o};\pm }>1$, the solution of the p-SLE (121) may be square-integrable only if it represents the PFSs near both endpoints, i.e., iff it satisfies the DBCs at ±1. Note that Everitt [59] defines the LP endpoint for the generic SLE via the requirement that the symplectic form

$$[{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}],{\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]]:=(1-{\mathsf{\eta }}^2)\mathrm{W}\{{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}],{\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\}$$

(131)

for any pair of the square-integrable solutions of the SLE (116) vanishes at ±1. This requirement automatically holds if both solutions obey the DBCs, and, therefore, this is true for any pair of the square-integrable solutions if ${\mathsf{\lambda }}_{\mathrm{o};\pm }>1$.

If one of the endpoints is LC, i.e., if $0<{\mathsf{\lambda }}_{\mathrm{o};+}<1$ or $0<{\mathsf{\lambda }}_{\mathrm{o};-}<1$, then the suggested algorithm selects only the square-integrable solutions, which constitute the PFSs near both quantization ends. We have already run into these solutions in Section 4 while discussing the RRZTs converting the LP region of the restr-HRef CSLE (35) into the LC region of the JRef CSLE (1).

Note that our approach does not allow one to formulate the spectral problem for the Jacobi differential equation if either −1 < α ≤ 0 or −1 < β ≤ 0 [59]. As discussed below, this limitation might affect the analysis of the transformed solutions in the LC region.

By analogy with the analysis presented above, the restr-HRef CSLE (35) for σ = − can be represented in the following prime form:

$$\begin{gathered} \{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}(1-{\mathsf{\eta }}^2){\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}-\cancel{\text{q}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]+\mathsf{\epsilon }\}\cancel{\mathsf{\Psi }}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\mathsf{\epsilon }|-]=0 \\ \mathrm{f}\mathrm{o}\mathrm{r} \mathsf{\eta }\in (-1,+1), \end{gathered}$$

(132)

where

$$\cancel{\text{q}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]=({\mathsf{\eta }}^2-1){\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]+{(1-{\mathsf{\eta }}^2)}^{-1}\mathrm{f}\mathrm{o}\mathrm{r} \mathsf{\eta }\in (-1,+1).$$

(133)

Keeping in mind that

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

(134)

we can again 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};\pm }$ while differing by their sign, and, therefore, one can determine the PFS by solving the p-SLE (132) under the DBCs

$$\underset{\mathsf{\eta }\to \pm 1}{lim}\cancel{\mathsf{\Psi }}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};{\tilde{\mathsf{\epsilon }}}_{\mathrm{n}}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})|-]=0$$

(135)

Returning to the special case (88), we choose

$${\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }={\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}$$

(136)

and

$${\mathsf{\lambda }}_{\pm }=\mp *{\mathsf{\lambda }}_{\pm }-1$$

(137)

under the restriction

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

(138)

requiring both endpoints of the JRef CSLE to be LP singularities. The vector parameters (61) are defined via the following relations:

$$*{\mathsf{\lambda }}_{\mp }={\mathsf{\lambda }}_{\mathrm{o};\mp }\pm 1=\pm {\mathsf{\lambda }}_{\mp }\pm 1=*{\mathsf{\lambda }}_{\mathrm{o};\mp }>0$$

(139)

It directly follows from (138) that $\stackrel{\rightharpoonup }{\mathsf{\sigma }}=+-$ and, hence, $\mathsf{\sigma }=-$, as expected.

Converting the solution (69) of the given CSLE to its prime form via the gauge transformation

$${\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]=\sqrt{1-{\mathsf{\eta }}^2}\mathsf{\varphi }[\mathsf{\eta };*{\mathsf{\lambda }}_{-}-1,-*{\mathsf{\lambda }}_{+}-1|1;+-,\mathrm{j}],$$

(140)

one can directly verify that the RRZ$\text{T}$ of the j-th eigenfunction of the p-SLE (121),

$${\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]={\displaystyle \frac{{\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]}{\mathsf{\eta }-b(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})}}{\stackrel{⌢}{\mathrm{P}}}_{\mathrm{j}+1}^{(*{\mathsf{\lambda }}_{\mathrm{o};+},*{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta }) \mathrm{f}\mathrm{o}\mathrm{r} \mathsf{\eta }\in [-1,+1]$$

(141)

satisfies the DBCs at the ends of the specified interval and therefore represents an eigenfunction of the transformed p-SLE (132) with the eigenvalue ${\mathsf{\epsilon }}_{\mathrm{j}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})$. Again, the eigenfunctions are squarely integrable:

$${\displaystyle \underset{-1}{\overset{+1}{\int }}\mathrm{d}\mathsf{\eta }}{\cancel{\mathsf{\psi }}}_{\mathrm{j}}^2[\mathsf{\eta };\text{*}{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]<\infty$$

(142)

and mutually orthogonal:

$${\displaystyle \underset{-1}{\overset{+1}{\int }}\mathrm{d}\mathsf{\eta }}{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };\text{*}{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]{\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };\text{*}{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]=0(\mathrm{j}\ne {\mathrm{j}}^{\prime }),$$

(143)

which brings us to the renowned orthogonality relations [15] for the X₁-Jacobi OPS

$${\displaystyle \underset{-1}{\overset{+1}{\int }}\mathrm{d}\mathsf{\eta }}{\stackrel{⌢}{\mathsf{\omega }}}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]{\stackrel{⌢}{\mathrm{P}}}_{\mathrm{n}}^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta }){\stackrel{⌢}{\mathrm{P}}}_{{\mathrm{n}}^{\prime }}^{(\mathsf{\alpha },\mathsf{\beta })}(\mathsf{\eta })=0(\mathsf{\alpha },\mathsf{\beta }>0;\mathrm{n}\ne {\mathrm{n}}^{\prime })$$

(144)

with the weight (78). The sequence starts from the eigenfunction ${\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]$, which is necessarily nodeless, since the first-degree polynomial (92) has the zero inside the infinite interval (−∞,−1):

$$c(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}):=-{\displaystyle \frac{*{\mathsf{\lambda }}_{\mathrm{o};-}+*{\mathsf{\lambda }}_{\mathrm{o};+}+2}{*{\mathsf{\lambda }}_{\mathrm{o};+}-*{\mathsf{\lambda }}_{\mathrm{o};-}}}<-1$$

(145)

under the constraint (39). This implies the Dirichlet problem of our interest may not have solutions at energies smaller than ${\mathsf{\epsilon }}_0({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})$. It directly follows from Proposition 5.1 in [15] that the X₁-Jacobi polynomial of degree n has exactly n − 1 zeros in (−1,+1), and, therefore, no solution may exist between two sequential eigenvalues ${\mathsf{\lambda }}_{\mathrm{k}-1}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})$ and ${\mathsf{\lambda }}_{\mathrm{k}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})$. Since the given sequence of the eigenvalues is unrestricted from above, the q-RSs (141) form the complete set of the eigenfunctions for the given Dirichlet problem.

Note that the symplectic form for two eigenfunctions [59],

$$\begin{gathered} [{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-],{\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]]:= \\ (1-{\mathsf{\eta }}^2)\mathrm{W}\{\{{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-],{\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]\}\mathrm{for}|\mathsf{\eta }|<1 \end{gathered}$$

(146)

vanishes at both singular ends ±1, and, therefore, it must vanish for any two eigenfunctions of the generic SLE

$$\{-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}\text{p}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}+{\text{q}}^{\text{p}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]-{\mathsf{\epsilon }}_{\mathrm{j}}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})w^{\text{p}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\}{\mathsf{\psi }}_{\mathrm{j}}^{\text{p}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]=0$$

(147)

with the canonical form (35), i.e., iff

$$\text{p}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\equiv {\mathrm{w}}^{\text{p}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}](1-{\mathsf{\eta }}^2)>0$$

(148)

and

$${\text{q}}^{\text{p}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]=-\text{p}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-]+\mathrm{I}\{\text{p}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\}$$

(149)

In particular, Everitt [60] chose

$$\text{p}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]={\text{p}}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]={(1-\mathsf{\eta })}^2{\mathsf{\omega }}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]$$

(150)

which brought him to the boundary conditions

$${\text{p}}_{\mathsf{\alpha },\mathsf{\beta }}[\mathsf{\eta }]\mathrm{W}{\{{\stackrel{⌢}{\mathrm{P}}}_1^{(*{\mathsf{\lambda }}_{\mathrm{o};+},*{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta }),{\stackrel{⌢}{\mathrm{P}}}_{\mathrm{n}}^{(*{\mathsf{\lambda }}_{\mathrm{o};+},*{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta })\}|}_{\mathsf{\eta }=\pm 1}=0$$

(151)

proven in his Remark 3.1iii. Imposing the Dirichlet boundary conditions (135) on the solutions of the p-SLE (132) thus presents the alternative way to obey the boundary conditions (151) for Case 1 in [60]. Though his Case 2 (−1 < α,β < 0) is not covered by the technique developed here, the crucial advantage of our approach is that, as discussed in Section 6.2, it can be easily extended to the infinite interval [1,∞), despite the fact that the EOPs infinitely grow at the upper endpoint.

Before concluding the discussion of the X₁-Jacobi OPS, let us mention some new by-product potential developments associated with the extension of the p-SLE (132) to the complex field. Obviously, the q-RSs (141) of the complexified p-SLE obey the DBCs at ±1 if $Re*{\mathsf{\lambda }}_{\mathrm{o};\pm }>0$. As a result, the complexified symplectic form (146) preserves its zero values at these points. We conclude that the q-RSs constructed in such a way are orthogonal with the complex weight (78) on the interval [−1,+1], assuming that the third pole lies outside the real interval [−1,+1]. Since the given orthogonality weight is complex, we refer to the corresponding complex polynomials in a real argument as being ‘quasi-orthogonal’. The discovery of the infinite quasi-orthogonal sequences of the complex X₁-Jacobi polynomials certainly goes beyond the conventional theory of the X-Jacobi OPSs. In Section 8, we will discuss similar finite quasi-orthogonal polynomial sequences defined on the infinite interval [1,∞).

6.2. Finite Orthogonal Subsequences of X₁-Jacobi DPS

We finally switch our discussion to the subject that has received very little attention compared with the X₁-Jacobi OPS [7,8,15,45,54,56,57,60,61,62]. Namely, our next step is to formulate the Sturm–Liouville problem on the infinite interval [1,∞) first for the JRef CSLE (1) and then for the restr-HRef CSLEs (35), taking advantage of the fact that the extraneous pole of the latter CSLE lies outside the cited interval in both cases σ = ±, provided that the ExpDiffs for the poles at ±1 are restrained by the condition (39) for σ = −.

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

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

(152)

which brings us to the p-SLE [57]

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

(153)

with the zero-energy free term

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

(154)

and the positive weight

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

(155)

Taking into account that

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

(156)

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. In contrast to the poles in the finite plane, the ExpDiff for the pole of the JRef CSLE (1) at infinity turns out to be energy-dependent. Keeping in mind that

$$\underset{\mathsf{\eta }\to \infty }{lim}[(\mathsf{\eta }-1)\cancel{\text{q}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]]=0$$

(157)

we find 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 }}$ while differing from each other by the sign. We thus confirm that the PFSs of the p-SLE (153) near both singular endpoints) are unambiguously determined by the DBCs

$${\cancel{\mathsf{\psi }}}_{\mathrm{j}}[1;*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=\underset{\mathsf{\eta }\to \infty }{lim}{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=0,$$

(158)

as asserted.

One can directly verify that the q-RSs

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

(159)

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

(160)

associated with the eigenvalues

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

(161)

satisfy the DBCs (158) for

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

(162)

and therefore are normalizable with the weight (155):

$${\displaystyle \underset{1}{\overset{\infty }{\int }}\mathrm{d}\mathsf{\eta }}{\cancel{\mathsf{\psi }}}_{-+,\mathrm{j}}^2[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]<\infty .$$

(163)

As the eigensolutions of the Sturm–Liouville problem solved under the DBCs, they must also be mutually orthogonal [29] with the weight (155):

$$\begin{gathered} {\displaystyle \underset{1}{\overset{\infty }{\int }}\mathrm{d}\mathsf{\eta }}{\cancel{\mathsf{\psi }}}_{-+,\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]{\cancel{\mathsf{\psi }}}_{-+,{\mathrm{j}}^{\prime }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=0 \\ \mathrm{f}\mathrm{o}\mathrm{r} {\mathrm{j}}^{\prime }<\mathrm{j}<{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};-},-{\mathsf{\lambda }}_{\mathrm{o};+}-1), \end{gathered}$$

(164)

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

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

(165)

with the weight

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

(166)

under constraint α > 0, β < 0, where we adopted Askey’s [31] definition of the R-Jacobi polynomials which, as proven by Chen and Srivastava [34], is equivalent to the elementary formula

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

(167)

with

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

(168)

Note that we [11] (see also [63]) changed the symbol R for J to avoid the confusion with Romanovski/pseudo-Jacobi [32,33] polynomials (R-Routh polynomials in our terms [53]) denoted in the recent publications [64,65,66,67,68] by the same letter ‘R’. A certain disarray may come from the fact that Koepf and Masjed-Jamei [35,69] used the symbol J for the R-Routh polynomials, which is inconsistent with the polynomial names in our classification scheme of the Romanovsi polynomials [30]. Note also that the symbol $\Re$ used for the R-Jacobi polynomials in the recent paper [70] was reserved by us [53] for the Routh polynomials [51]

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

While R-Routh polynomials form finite orthogonal subsequences of the Routh DPS, the R-Jacobi polynomials were discovered by Romanovski [30] and have no relation to Routh’ work, contrary to the statement made in [71].

Comparing (167) with (2.8) in [72] shows that the Majed-Jamei’s M-polynomials are related to the R-Jacobi polynomials (167) via the elementary formula

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

(169)

(It is worth mentioning that the abbreviation ‘OPS’ in [72] stands for the orthogonal polynomial but not for the infinite ‘orthogonal polynomial system’ broadly abbreviated as ‘OPS’ in the modern theory of the exceptional orthogonal polynomials [15,16,17,22,23,45].)

Substituting q = α, p = −α − β into (2.14) in [72] gives

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

(170)

As anticipated from (169), the corresponding formula for the R-Jacobi polynomials (see (5) in [73], with $\mathsf{\theta },\mathsf{\vartheta }$ standing for α, β here) differs by the factor (n!)². Indeed, keeping in mind that

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

one can directly verify that the integral (170) differs exactly by the factor (n!)² from Askey’s [31] expression for the square integrals of the R-Jacobi polynomials.

The main advantage of formulating the Dirichlet problem for the prime SLE (153) rather than for its canonical form (1) solved on the infinite interval [1,∞) is that the DBCs can be implied within both LP (${\mathsf{\lambda }}_{\mathrm{o};+}>1$) and LC ($0<{\mathsf{\lambda }}_{\mathrm{o};+}<1$) ranges of the ExpDiff for the pole of the SLE at +1. On the contrary, the JRef CSLE (1) can be solved under the mentioned DBCs only if the pole at +1 lies in the LP region, i.e., exclusively in the parameter domain, where the eigenfunctions of the CSLE on the infinite interval [1,∞) are unambiguously determined by the requirement to be square-integrable with the weight (2) for η > 1.

One still needs to prove that the q-RSs (160) form the complete set of the eigenfunctions of the given Dirichlet problem. This can be performed, for example, by converting the JRef CSLE (1) to the hypergeometric equation, following the arguments presented by the author [2] for the exactly solvable JRef CSLE with the properly chosen density function.

Similarly to the setback of our technique for the classical Jacobi polynomials, our current 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 $\mathrm{R}\mathrm{R}\mathrm{Z}\text{T}\mathrm{s}$ of the R-Jacobi polynomials discussed below.

Since the p-SLE (153) has the discrete energy spectrum only for ${\mathsf{\lambda }}_{\mathrm{o};-}>{\mathsf{\lambda }}_{\mathrm{o};+}+1,$ examination of the constraint (39) reveals that the SLE (35) does not have normalizable solutions on the interval [1,∞) if $\mathsf{\sigma }=-$. Starting from this point, we will consider the DBC at infinity only for the SLE (35) with $\mathsf{\sigma }=+.$ Before coming to the case of our current interest (m = 1), let us note that all the zeros of the solutions (127) lie between −1 and +1 for any j > 0, and, therefore, any of them can be used as the TF for the RRZT [11]. Each solution represents the PFS near the lower singular end point of the interval [1,∞) at the energy

$$\begin{gathered} {\underset{˜}{\mathsf{\epsilon }}}_{++,\mathrm{m}}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})=-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{({\mathsf{\lambda }}_{\mathrm{o};-}+{\mathsf{\lambda }}_{\mathrm{o};+}+2\mathrm{m}+1)}^2<{\underset{˜}{\mathsf{\epsilon }}}_{-+,0}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{m}\ge 1 \\ (\mathsf{\sigma }=+). \end{gathered}$$

(171)

In this paper, we make use of the very first solution in this infinite sequence of the TFs formed by classical Jacobi polynomials. Alternatively, we can consider the RRZT with the TF represented by the PFS near infinity at the energy

$${\underset{˜}{\mathsf{\epsilon }}}_{--,1}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})=-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{({\mathsf{\lambda }}_{\mathrm{o};-}+{\mathsf{\lambda }}_{\mathrm{o};+}-3)}^2<{\underset{˜}{\mathsf{\epsilon }}}_{-+,0}({\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})$$

(172)

which leads us to the same CSLE (35) with $\mathsf{\sigma }=+$. This is the direct manifestation of the form-invariance of the latter CSLE under the second-order RDCT with the seed functions $*{\mathsf{\varphi }}_{++,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]$ and ${\mathsf{\varphi }}_{--,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]$.

To formulate the Sturm–Liouville problem on the infinite interval [1,∞) for the restr-HRef CSLE with $\mathsf{\sigma }=+,$ we convert the latter to its prime form with the leading coefficient function (152) and weight (155):

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

(173)

where the zero-energy free term has the generic form

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

(174)

Taking into account that

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

(175)

and

$$\underset{\mathsf{\eta }\to \infty }{lim}[(\mathsf{\eta }-1)\cancel{\text{q}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+]]=0$$

(176)

we confirm that the ChExps of the Frobenius solutions for the pole at +1 or at infinity have the same absolute value ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}*{\mathsf{\lambda }}_{\mathrm{o};+}$ or, respectively, ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{-\mathsf{\epsilon }}$ while differing by their sign in both cases. Since the PFS of the p-SLE (173) at the given energy is unambiguously selected by the DBC at the corresponding endpoint, the DBCs

$${\cancel{\mathsf{\psi }}}_{\mathrm{j}}[1;*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+]=\underset{\mathsf{\eta }\to \infty }{lim}{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+]=0$$

(177)

unequivocally determine all the eigenfunctions of the given SLP.

As illuminated below, re-formulating the spectral theory for the restr-HRef CSLE quantized on the infinite interval [1,∞) as the Dirichlet problem for the p-SLE (132) yields the simple but rigorous way to determine the quasi-rational eigenfunctions with the polynomial components forming the finite EOP sequence, taking advantage of the powerful spectral theorems proven by Gesztesy et al. [29] for solutions of the generic SLEs solved under the DBCs.

By applying the RRZT with the TF ${\cancel{\mathsf{\psi }}}_{++,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]$ to the eigenfunction (159), we find that the solution of the p-SLE (173)

$$\cancel{\mathsf{\psi }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|1;-+,\mathrm{j}]\propto {\displaystyle \frac{{\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };*\stackrel{\rightharpoonup }{\mathsf{\lambda }}]}{\mathsf{\eta }-\mathrm{b}(*\stackrel{\rightharpoonup }{\mathsf{\lambda }})}}{\mathrm{P}}_{\mathrm{j}+1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|1;-+,\mathrm{j}],$$

(178)

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

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

(179)

$$*{\mathsf{\lambda }}_{\mp }=\mp {\mathsf{\lambda }}_{\mathrm{o};\mp }\mp 1(*\mathsf{\sigma }=-),$$

(180)

and

$$*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}={\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}+1,$$

(181)

vanishes at η = +1 or, in other words, represents the PFS of the p-SLE (173) near the lower endpoint. Under the constraint (162), it also satisfies the DBC at infinity and therefore represents an eigenfunction of the p-SLE (173).

Theorem 4. 

The$\mathrm{R}\mathrm{R}\mathrm{Z}\text{T}\mathrm{s}$of the R-Jacobi polynomials form a finite EOP subset of the X-DPS of series J.

Proof. 

One can directly verify that the eigenfunctions constructed in such a way are normalizable with the weight (155):

$${\displaystyle \underset{1}{\overset{\infty }{\int }}\mathrm{d}\mathsf{\eta }}{\cancel{\mathsf{\psi }}}_{\mathrm{j}}^2[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|1;-+,\mathrm{j}]\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]<\infty \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}<{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};-},-{\mathsf{\lambda }}_{\mathrm{o};+}-1).$$

(182)

Again, as the eigensolutions of the Sturm–Liouville problem solved under the DBCs, they must be mutually orthogonal [29] with the same weight on the infinite interval in question:

$$\begin{gathered} {\displaystyle \underset{1}{\overset{\infty }{\int }}\mathrm{d}\mathsf{\eta }}{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|1;-+,\mathrm{j}]{\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}^{\prime }|1;-+,\mathrm{j}]\cancel{w}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=0 \\ \mathrm{f}\mathrm{o}\mathrm{r} {\mathrm{j}}^{\prime }<\mathrm{j}<{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};-},-{\mathsf{\lambda }}_{\mathrm{o};+}-1). \end{gathered}$$

(183)

Combining (180) and (181), we can re-write (178) and the associated eigenvalues as

$${\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };-*{\mathsf{\lambda }}_{\mathrm{o};-},*{\mathsf{\lambda }}_{\mathrm{o};+}|-+,1]={\displaystyle \frac{{\cancel{\mathsf{\psi }}}_0[\mathsf{\eta };-*{\mathsf{\lambda }}_{\mathrm{o};-},*{\mathsf{\lambda }}_{\mathrm{o};+}]}{\mathsf{\eta }-b(-*{\mathsf{\lambda }}_{\mathrm{o};-},*{\mathsf{\lambda }}_{\mathrm{o};+})}}{\stackrel{⌢}{\mathrm{P}}}_{\mathrm{j}+1}^{(*{\mathsf{\lambda }}_{\mathrm{o};+},-*{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta })$$

(184)

and

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

(185)

respectively. It then directly follows from (183) that the X₁-Jacobi polynomials in question are mutually orthogonal with the weight

$$\mathrm{W}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|1,-+]:={\displaystyle \frac{4{(1+\mathsf{\eta })}^{-*{\mathsf{\lambda }}_{\mathrm{o};-}}{(1-\mathsf{\eta })}^{*{\mathsf{\lambda }}_{\mathrm{o};+}}}{{(*{\mathsf{\lambda }}_{\mathrm{o};-}+*{\mathsf{\lambda }}_{\mathrm{o};+})}^2|\mathsf{\eta }-b(-*{\mathsf{\lambda }}_{\mathrm{o};-},*{\mathsf{\lambda }}_{\mathrm{o};+}){|}^2}}$$

(186)

namely,

$$\begin{gathered} {\displaystyle \underset{1}{\overset{\infty }{\int }}\mathrm{d}\mathsf{\eta }}\mathrm{W}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|1,-+]{\stackrel{⌢}{\mathrm{P}}}_{\mathrm{n}}^{(*{\mathsf{\lambda }}_{\mathrm{o};+},-*{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta }){\stackrel{⌢}{\mathrm{P}}}_{{\mathrm{n}}^{\prime }}^{(*{\mathsf{\lambda }}_{\mathrm{o};+},-*{\mathsf{\lambda }}_{\mathrm{o};-})}(\mathsf{\eta })=0 \\ \mathrm{f}\mathrm{o}\mathrm{r} 1\le {\mathrm{n}}^{\prime }<\mathrm{n}<{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}({\mathsf{\lambda }}_{\mathrm{o};-},-{\mathsf{\lambda }}_{\mathrm{o};+}-1), \end{gathered}$$

(187)

which completes the proof of Theorem 4. □

Theorem 4 constitutes one of the most important issues addressed by this analysis. Namely, we proved that at least some eigenfunctions of the p-SLE (173) solved under the DBCs are expressible in terms of a finite subset of the X₁-DPS of type J. Yadav et al. [37] termed this subset simply as ‘X₁-Jacobi polynomials’ (with reference to [15]), without going into any details. The surprising new element of their use of this term is that the indexes of the X₁-Jacobi polynomials in question have opposite signs and therefore do not belong to the X₁-OPS. The cited assertion in [37] stimulated my interest in this problem [38], giving rise to the concept of the X-Jacobi DPSs, which may contain both X-Jacobi OPSs [22,23] and finite EOP sequences formed by the $\mathrm{R}\mathrm{D}\mathrm{C}\text{T}\mathrm{s}$ of the R-Jacobi polynomials.

While the X₁-Jacobi OPS has been scrupulously studied in [15,41,54], the finite X₁-Jacobi EOPs avoided mathematicians’ attention so far. The proof of their existence in the LC region of the restr-HRef CSLE (35) provides an important new step in developing the precise mathematical theory of finite EOP sequences. The important novel element of our analysis is that the RDTs of our interest convert each PFS near the pole at +1 into the corresponding PFS of the transformed CSLE. The formulated Dirichlet problem thus allows us to single out the PFSs of the p-SLE from their LC background composed of all other normalizable eigenfunctions. The polynomial components of the q-RSs vanishing at both singular endpoints form the sought-for EOP sequence.

Similarly to the Dirichlet problem formulated for the R-Jacobi polynomials, the advantage of using the prime SLE (173), instead of its canonical form (135), comes from the fact that the OBCs can be imposed within both LP ($*{\mathsf{\lambda }}_{\mathrm{o};+}>1$) and LC ($0<*{\mathsf{\lambda }}_{\mathrm{o};+}<1$) ranges of the ExpDiff for the pole of the SLE at +1. On the contrary, the restr-HRef CSLE (135) can be solved under the DBCs only if the pole at +1 lies in the LP region, i.e., exclusively in the domain where the eigenfunctions sought for on the infinite interval [1,∞) are unambiguously determined by the requirement to be square-integrable with the weight (2) for η > 1.

Let us now demonstrate the power of the developed formalism by proving that the q-RSs (178) form the complete set of the eigenfunctions for the formulated Dirichlet problem—the issue completely ignored in the literature [9,10,37]. As the starting point, we first need to confirm that the polynomial component of the q-RS (178) has exactly j real zeros larger than 1.

Proposition 1. 

The polynomial of degree j + 1 in the given EOP sequence has one real zero between −1 and 1 and j real zeros larger than 1.

Proof. 

First note that the EOP sequence in question starts from the first-degree polynomial (92), having a zero at

$$c(*\stackrel{\rightharpoonup }{\mathsf{\lambda }}):={\displaystyle \frac{*{\mathsf{\lambda }}_{-}+*{\mathsf{\lambda }}_{+}+2}{*{\mathsf{\lambda }}_{-}-*{\mathsf{\lambda }}_{+}}}$$

(188)

or, making use of (180) and (181),

$$c(-*{\mathsf{\lambda }}_{\mathrm{o};-},*{\mathsf{\lambda }}_{\mathrm{o};+})={\displaystyle \frac{*{\mathsf{\lambda }}_{\mathrm{o};-}-*{\mathsf{\lambda }}_{\mathrm{o};+}-2}{*{\mathsf{\lambda }}_{\mathrm{o};-}+*{\mathsf{\lambda }}_{\mathrm{o};+}}}=1-{\displaystyle \frac{2(*{\mathsf{\lambda }}_{\mathrm{o};+}+1)}{*{\mathsf{\lambda }}_{\mathrm{o};-}+*{\mathsf{\lambda }}_{\mathrm{o};+}}}<1.$$

(189)

It then directly follows from the orthogonality relations (187) that the polynomial of the degree j + 1 must have at least j zeros larger than 1. □

However, we still need to prove that no polynomial of degree j+1 may have all the zeros lying within the interval (1,∞) or, to be more precise, that each polynomial in the sequence has a single zero inside the interval (−∞,1). The proof directly follows from our observation that the X₁-Jacobi polynomial of degree j + 1 coincides (after being converted to its monic form) with the second polynomial in the sequence of the monic X_j+1-Jacobi polynomials with the same indexes. Namely, re-writing (A26), coupled with (A29), as

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

(190)

and making use of (A23) for $\stackrel{\rightharpoonup }{\cancel{\mathsf{\sigma }}}=-+$ shows that

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

(191)

The crucial point for our current discussion is that both indexes of the polynomial on the right are positive for ${{\mathsf{\lambda }}^{\prime }}_{+}={\mathsf{\lambda }}_{\mathrm{o};+}>1$ and ${{\mathsf{\lambda }}^{\prime }}_{-}<0$, which implies that the polynomial (191) in the reflected argument -η belong to the X_j-Jacobi OPS (in the terminology of [45,46]) and, in particular,

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

(192)

as asserted. It then directly follows from Proposition 5.4 in [46], with m = 1, that each polynomial (192) under the constraint (162) has one zero between −1 and +1, and, therefore, the polynomial of order j + 1 may not have more than j zeros larger than 1. This completes the proof of Proposition 1.□

We are now ready to prove that the q-RSs (174) represent all the possible solutions of the Dirichlet problem of our interest.

Theorem 5. 

The Dirichlet problem for the p-SLE (163) defined on the infinite positive interval [1,∞) does not have any solutions other than the eigenfunctions (174), assuming that the pole of the corresponding CSLE (35) at +1 is LP.

Proof. 

Since the very first q-RS in the sequence (174) does not have nodes in [1,∞), the Dirichlet problem in question may not have solutions below the energy ${\underset{˜}{\mathsf{\epsilon }}}_0(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-+)$. As the direct consequence of Proposition 1, we also assert that there may be no eigenfunctions between any two sequential eigenvalues ${\underset{˜}{\mathsf{\epsilon }}}_{\mathrm{n}-1}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-+)$ and ${\underset{˜}{\mathsf{\epsilon }}}_{\mathrm{n}}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|-+).$ □

However, compared with the proof presented by us for the X₁-Jacobi OPS, a certain complication comes from the fact that there is only a finite number of the eigenfunctions expressible in terms of X₁-Jacobi polynomials. Therefore, we also need to confirm that no negative eigenvalue exists above the largest eigenvalue in the sequence (185) under the constraint (162).

Suppose that the given Dirichlet problem has a solution ${\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+]$ at an energy ${\underset{˜}{\mathsf{\epsilon }}}_{\mathrm{n}}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+)$. The RRZT with the TF $*{\mathsf{\varphi }}_{++,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]$ then converts the latter eigenfunction into the following q-RS of the p-SLE (153):

$$\cancel{\mathsf{\psi }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]={\cancel{\mathsf{\psi }}}_{++,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\mathrm{W}\{\sqrt{{\mathsf{\eta }}^2-1}/{\cancel{\mathsf{\psi }}}_{++,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}],{\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+]\}$$

(193)

$$=-{\displaystyle \frac{{\mathsf{\eta }}^2-1}{{\cancel{\mathsf{\psi }}}_{++,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}{\displaystyle \frac{{\cancel{\mathsf{\psi }}}_{++,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]{\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+]}{\sqrt{{\mathsf{\eta }}^2-1}}}$$

(194)

Examination of the sum

$$\begin{gathered} \cancel{\mathsf{\psi }}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]=-({\mathsf{\eta }}^2-1){\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}{\displaystyle \frac{{\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+]}{\sqrt{{\mathsf{\eta }}^2-1}}} \\ +\sqrt{{\mathsf{\eta }}^2-1}ld{\cancel{\mathsf{\psi }}}_{++,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]{\cancel{\mathsf{\psi }}}_{\mathrm{n}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+] \end{gathered}$$

(195)

reveals that each summand vanishes at the limit η → 1+, assuming that the pole the CSLE (35) at +1 lies in the LP region ($*{\mathsf{\lambda }}_{\mathrm{o};+}>1$).

Next, the derivative appearing in the first summand diminishes at infinity faster than η⁻², and, therefore, the summand vanishes in the limit η → ∞. Furthermore, keeping in mind that ${\cancel{\mathsf{\psi }}}_{++,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]$ is a quasi-rational function, one finds

$$\underset{\mathsf{\eta }\to \infty }{lim}\sqrt{{\mathsf{\eta }}^2-1}|ld{\cancel{\mathsf{\psi }}}_{++,1}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]|<\infty$$

(196)

which assures that the second summand in (195) also vanishes at infinity. We conclude that the q-RS (194) of the p-SLE (153) satisfies the DBCs (156) and (157) and therefore represents an eigenfunction with an eigenvalue differing from any of eigenvalues (161). This result contradicts the assertion that the latter represents the complete discrete energy spectrum of the given Dirichlet problem. We thus confirmed that the p-SLE (173) solved under the DBCs (175) and (176) may not have eigenfunctions other than (184). □

7. Liouville Potentials Shape-Invariant Under Second-Order RDCTs

In the preceding sections, we developed the rigorous theory of the SLPs solvable in terms of the infinite and finite EOP sequences composed of polynomial solutions of the Heun equation (94). The exactly solvable Liouville potentials for these problems were discovered in two separate pioneering works [7,9] by Quesne et al., using the heuristic prescriptions of the SUSY quantum mechanics. More recently, Soltész et al. [10] discussed the construction of these potentials in a unified matter but, again, within the SUSY framework (ignoring the more precise analysis of this problem in [12,38]).

The cornerstone of the developed technique is our observation [26,28] that the RZTs [24] turn into the conventional DTs after the CSLE under consideration of being converted by the Liouville transformation to the corresponding Schrödinger equation. The developed theory of the rationally deformed SLEs allows one to accurately justify the heuristic rules of SUSY quantum mechanics in the important case of solvable-by-polynomials Liouville potentials. Namely, to relate our approach to the conventional quantum-mechanical applications, we convert the CSLEs (1) and (35) by the gauge transformations

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

(197)

and, respectively,

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

(198)

to the algebraic SLEs:

$$\begin{gathered} \left\{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}{\mathit{\rho }}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}+(\mathsf{\epsilon }-\mathrm{V}[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]){\mathit{\rho }}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]\right\}\mathsf{\Psi }[\mathsf{\eta };{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\mathsf{\epsilon }]=0 \\ \mathit{sgn}\mathsf{\epsilon }=\mathit{sgn}(1-\mathsf{\eta }) \end{gathered}$$

(199)

and

$$\begin{gathered} \left\{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}{\mathit{\rho }}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\eta }}}+(\mathsf{\epsilon }-\mathrm{V}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|\mathsf{\sigma }]){\mathit{\rho }}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]\right\}\mathsf{\Psi }[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}};\mathsf{\epsilon }|\mathsf{\sigma }]=0 \\ \mathit{sgn}\mathsf{\epsilon }=\mathit{sgn}(1-\mathsf{\eta })=-\mathsf{\sigma } \end{gathered}$$

(200)

with the rational Liouville potentials defined via the general formula (118) with

$$\cancel{\text{p}}[\mathsf{\eta }]={\mathit{\rho }}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]$$

(201)

namely,

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

(202)

and

$$\begin{gathered} {\mathit{\rho }}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]\mathrm{V}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|\mathsf{\sigma }]:=-{\mathit{\rho }}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|\mathsf{\sigma }]+{\mathit{\rho }}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]I\{{\mathit{\rho }}^{-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}[\mathsf{\eta }]\} \\ \mathit{sgn}(\mathsf{\eta }-1)=\mathsf{\sigma }. \end{gathered}$$

(203)

The change of variable $\mathsf{\eta }(\mathrm{x})$ determined by the first-order ODE

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

(204)

converts each SLE into the Schrödinger equation with the corresponding potential given by the conventional formulas [74]:

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

(205)

and

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

(206)

where the Schwarzian derivative is related to the generic function (119) via the elementary formula

$${\{\mathsf{\eta }(\mathrm{x}),\mathrm{x}\}=-2\cancel{\text{p}}[\mathsf{\eta }(\mathrm{x})]I\{\cancel{\text{p}}[\mathsf{\eta }]\}|}_{\mathsf{\eta }=\mathsf{\eta }(\mathrm{x})}$$

(207)

Substituting (4) and

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

(208)

into the right-hand side of (202) gives

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

(209)

Setting

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

(210)

or

$$\begin{gathered} 2\mathrm{A}+1={\mathsf{\lambda }}_{\mathrm{o};-}-{\mathsf{\lambda }}_{\mathrm{o};+},2\mathrm{B}:={\mathsf{\lambda }}_{\mathrm{o};+}+{\mathsf{\lambda }}_{\mathrm{o};-}, \\ , \\ \mathsf{\alpha }:={\mathsf{\lambda }}_{\mathrm{o};+}\equiv \mathrm{B}-\mathrm{A}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\mathsf{\beta }:=-{\mathsf{\lambda }}_{\mathrm{o};-}\equiv -\mathrm{A}-\mathrm{B}-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\hspace{1em}\mathrm{for}\mathsf{\eta }>1 \end{gathered}$$

(211)

and making the change of variable

η(x) = sin x

(212)

or

η(x) = cosh x

(213)

on the intervals (−1,+1) or (1,∞) accordingly, we come to the Schrödinger equation with the ‘Scarf I’ potential in [8] and the ‘generalized PT’ potential in [9] (t-PT and h-PT potentials in our terms). On the other hand, representing (209) for |η| < 1 as

$$\mathrm{V}[\mathsf{\eta };\mathsf{\beta },\mathsf{\alpha }]={\displaystyle \frac{({\mathsf{\alpha }}^2-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.)(\mathsf{\eta }+1)-({\mathsf{\beta }}^2-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.)(\mathsf{\eta }-1)}{2(1-{\mathsf{\eta }}^2)}}$$

and making the change in variable

$$1-\mathsf{\eta }=2si{n}^2{\displaystyle \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

brings us to the already mentioned potential function (37) in [43] with the shifted zero-point energy and x replaced for x/2 in our notation.

Imposing the DBCs on the eigenfunctions of the Schrödinger equations with the t-PT [8] and h-PT [9] potentials is fully equivalent to the Dirichlet problems formulated for the algebraic SLE (189) on the intervals [−1,+1] and [1,∞), respectively. As a result the DBC at the lower quantization end of the interval [−1,+1] or [1,∞) unambiguously determines the PFS near the given singular endpoint iff ${\mathsf{\lambda }}_{\mathrm{o};-}>{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ or accordingly, i.e., iff A > B + 1 [8] and B < A + 1 [9] for the t- and h-PT potentials, respectively.

As pointed to [8] and correspondingly in [9], the latter constraints automatically assure that the corresponding potential is repulsive near the given end. By converting the JRef CSLE (1) to its prime forms (127) and (153) on the intervals [−1,+1] and [1,∞), respectively, we can study the spectral peculiarities of the potential reductions with infinitely deep wells at one of the singular end.

Also note that any solution of the algebraic Schrödinger equation (199) with the t-PT potential is normalizable on the finite interval [−1,+1] if ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}<{\mathsf{\lambda }}_{\mathrm{o};\mp }<1$. Similarly, any PFS of the latter equation near infinity is normalizable on the infinite interval [1,∞) if ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}<{\mathsf{\lambda }}_{\mathrm{o};+}<1$. In both cases, we thus deal with the regions where the physical potentials have continuously infinite manifolds of bound states.

The derived expressions for the t- and h-PT potentials turn into the unified formula (18) in [10] if we set

$$\mathrm{z}(\mathrm{x})\equiv \mathsf{\eta },\mathrm{C}=\mathit{sgn}(1-\mathsf{\eta }),\mathsf{\alpha }={\mathsf{\lambda }}_{\mathrm{o};+},\mathsf{\beta }=\mathrm{C}{\mathsf{\lambda }}_{\mathrm{o};-}$$

(214)

However, there are two lapses in Lévai et al.’s formula (19) describing the corresponding eigenfunctions. Namely, $1-\mathsf{\eta }$ must be changed for |$1-\mathsf{\eta }$|, and (what is even more important!) one must distinguish between the classical Jacobi polynomials for |η| < 1 and the R-Jacobi polynomials converted to the Jacobi form via (167). We shall come back to this issue after introducing a similar representation for the Liouville potentials of the SLEs (201).

Replacing ${\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}$ for $*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}$ in (202) and subtracting the resultant function from (203), one finds

$$\mathrm{V}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|\mathsf{\sigma }]=\mathrm{V}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]-{\mathit{\rho }}^{-1}[\mathsf{\eta }]\{{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|\mathsf{\sigma }]-{\mathrm{I}}^{\mathrm{o}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}]\}$$

(215)

or, making use of (36),

$$\begin{gathered} \mathrm{V}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|\mathsf{\sigma }]=\mathrm{V}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}] \\ -2\mathrm{sgn}(\mathsf{\eta }-1)\left\{{\displaystyle \frac{1}{\mathsf{\eta }/{\mathrm{e}}_3^{\mathsf{\sigma }}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})-1}}+{\displaystyle \frac{1-1/{[{\mathrm{e}}_3^{\mathsf{\sigma }}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})]}^2}{{[\mathsf{\eta }/{\mathrm{e}}_3^{\mathsf{\sigma }}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})-1]}^2}}\right\} \end{gathered}$$

(216)

under the restriction $\mathit{sgn}(\mathsf{\eta }-1)=\mathsf{\sigma }.$ Setting

$$\begin{gathered} 2\mathrm{A}-1:=*{\mathsf{\lambda }}_{\mathrm{o};+}+*{\mathsf{\lambda }}_{\mathrm{o};-},2\mathrm{B}:=*{\mathsf{\lambda }}_{\mathrm{o};-}-*{\mathsf{\lambda }}_{\mathrm{o};+}, \\ \mathsf{\alpha }:=*{\mathsf{\lambda }}_{\mathrm{o};+},\mathsf{\beta }:=*{\mathsf{\lambda }}_{\mathrm{o};-},{\mathrm{e}}_3^{-}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})={\displaystyle \frac{2\mathrm{A}-1}{2\mathrm{B}}}\hspace{1em}\mathrm{for}\left|\mathsf{\eta }\right|<1 \end{gathered}$$

(217)

or

$$\begin{gathered} 2\mathrm{A}+1:=*{\mathsf{\lambda }}_{\mathrm{o};-}-*{\mathsf{\lambda }}_{\mathrm{o};+}>0,2\mathrm{B}:=*{\mathsf{\lambda }}_{\mathrm{o};-}+*{\mathsf{\lambda }}_{\mathrm{o};+}, \\ \mathsf{\alpha }:=*{\mathsf{\lambda }}_{\mathrm{o};+},\mathsf{\beta }:=-*{\mathsf{\lambda }}_{\mathrm{o};-},{\mathrm{e}}_3^{+}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})={\displaystyle \frac{2\mathrm{A}+1}{2\mathrm{B}}}\hspace{1em};\mathrm{for}\mathsf{\eta }>1 \end{gathered}$$

(218)

and making the change in variable (212) or (213), respectively, brings us to the potential (3.5) in [8] or accordingly (9) in [9].

Again, setting

$$\mathrm{z}(\mathrm{x})\equiv \mathsf{\eta },\mathrm{C}=\mathit{sgn}(1-\mathsf{\eta }),\mathsf{\alpha }=*{\mathsf{\lambda }}_{\mathrm{o};+},\mathsf{\beta }=\mathrm{C}*{\mathsf{\lambda }}_{\mathrm{o};-},$$

(219)

we come to the unified formula (21) for these two potentials in [10] (Note that the title of the cited paper is misleading, since none of these potentials is shape-invariant in the conventional sense [5]).

Keeping in mind that

$$b^{-1}(-\mathsf{\beta },\mathsf{\alpha })=b(\mathsf{\beta },\mathsf{\alpha })$$

(220)

we conclude that

$${\mathrm{e}}_3^{\mathsf{\sigma }}(*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}})\equiv b(\mathsf{\beta },\mathsf{\alpha })$$

(221)

in both cases σ = + and σ = −. This important observation made by Lévai et al. [10] allowed these authors to represent the eigenfunctions of the Schrödinger equation with the potentials (216) in the uniform fashion, assuming that $1-\mathsf{\eta }$ in their Formula (22) is replaced for |$1-\mathsf{\eta }$|, and (what is crucial for the current analysis) one makes a clear distinction between the X-orthogonal polynomial components of the q-RSs appearing in the right-hand side of the cited formula. Namely, these polynomial components form the infinite X₁-Jacobi OPS [15,45] for Quesne’s [8] rationally deformed t-PT potential and the finite EOP sequence composed of the $\mathrm{R}\mathrm{D}\text{T}\mathrm{s}$ of R-Jacobi polynomials for the BQR potential [9]. This fundamentally important clarification of the term ‘X₁-Jacobi polynomials’ introduced by Yadav et al. [37] was thoroughly elucidated in my presentation in Prague [38], which was simply ignored in [10].

Again, imposing the DBCs on the eigenfunctions of the Schrödinger equation with the potentials (3.5) and (9) in [8,9], respectively, is fully equivalent to the Dirichlet problems formulated for the algebraic Schrödinger equations (200) with σ = − and σ = + on the intervals [−1,+1] and [1,∞) accordingly. As a result, the DBC at the lower quantization end of the interval [−1,+1] or [1,∞) unambiguously determines the PFS near this singular endpoint iff $*{\mathsf{\lambda }}_{\mathrm{o};-}>{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ or, accordingly, i.e., iff or B < A+1 [9] for the $\mathrm{R}\mathrm{D}\text{T}$ of the t- or h-PT potential, respectively. Again, the latter constraints automatically assure that the corresponding potential is repulsive near the given end [8,9].

By analogy with the t-PT potential, any solution of the algebraic Schrödinger equations (200) with σ = − is normalizable on the finite interval [−1,+1] if ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}<*{\mathsf{\lambda }}_{\mathrm{o};\mp }<1$. Similarly, any PFS of the algebraic Schrödinger equations (200) near infinity is normalizable on the infinite interval [1,∞) for σ = + if ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}<*{\mathsf{\lambda }}_{\mathrm{o};+}<1$.

Finally, let us notice that the parameter swap $\mathrm{B}\leftrightarrow \mathrm{A}+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ discussed in [75] is equivalent to the interchange in the values of the ExpDiffs $*{\mathsf{\lambda }}_{\mathrm{o};\mp }$. As expected, the so-called ‘new’ potential (14) in [75] has the pole on the infinite positive interval [1,∞), and, as a result, it cannot be analytically quantized. The mentioned parameter swap must be accompanied by the change in variable $cos\mathrm{x}\to -cos\mathrm{x}$, which is associated with the quantization of the restr-HRef CSLE for σ = + on the infinite negative interval (−∞, −1]. The resultant eigenfunctions thus become expressible in terms of the X₁-Jacobi polynomials in the reflected argument $-\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{x}$.

8. Discussion

One of the most important elements of the presented analysis is the use of RRZTs as a systematic tool for constructing new RCSLEs that are solvable in terms of quasi-rational functions. By converting each of the constructed RCSLEs to its ‘prime’ form [28], we re-formulated the associated Sturm–Liouville problem as the Dirichlet problem and then took advantage of the meticulous theorems proven by Gesztesy et al. [29] for eigenfunctions of generic SLEs solvable under DBCs. The advanced formalism put forward in [28] represents the valuable alternative to formulating the ‘rational SLP’ (RSLP) with the eigenpolynomials forming the X_m-Jacobi OPS [15,45,60] (see also [22] for further developments). In particular, the developed technique is applicable to the SLPs solved by both infinite and finite EOP sequences [11], in contrast to more familiar eigenoperators with the eigenpolynomials forming the X_m-Jacobi OPS [45,46,76]. While the SLP solvable via the X₁-Jacobi OPS was scrupulously analyzed by Everitt [60], its counterpart on the infinite interval [1,∞) has mostly escaped the attention of mathematicians.

By requiring the eigenfunctions of each prime SLEs to satisfy the DBCs, we accurately confirmed that the given $\mathrm{R}\mathrm{R}\mathrm{Z}\text{T}\mathrm{s}$ of the exactly solvable JRef CSLE (1) are themselves exactly solvable in terms of infinite or finite EOP sequences. The presented analysis is in sharp contrast with the numerous discussions of this issue in connection with the rational extensions of the t- and h-PT potentials [7,8,9,10,41,77,78,79], where the exact solvability of the corresponding Schrödinger equation in terms of X-orthogonal polynomials was usually taken for granted.

The developed formalism was used to provide the rigorous mathematical basis for the unified description [10] of eigenfunctions of the Schrödinger equation with the Liouville potentials (216) solvable by infinite (σ = −) and finite (σ = +) EOP sequences. It was demonstrated that the finite subsequence of X-orthogonal polynomials that was discovered in [9] is composed of the $\mathrm{R}\mathrm{D}\text{T}\mathrm{s}$ of the R-Jacobi polynomials.

The suggested approach also provides the accurate mathematical basis for the conventional SUSY theory of rationally extended potentials utilized in [7,8,9,10]. It should be stressed that the renowned SUSY rules [5] for the changes in energy spectra due to DTs of the Schrödinger equation were originally formulated [80,81] for potentials with exponential tails at ±∞. The latter restriction forced the ODE under consideration to have the LP singularities at the quantization ends. It has been then extended by Sukumar [82] to radial potentials without properly treating the LC range of the centrifugal barrier. Since then, the SUSY rules for the radial Schrödinger equation were copied without the attentive examination of this non-trivial problem, with Gangopadhyaya et al.’s paper [83] and Chapter 12 in [84] as the only known-to-me exceptions.

In [28], we performed the scrupulous analysis of the RRZTs of RCSLEs in the case when the ExpDiffs for the poles at the origin lie within the LC range for either original or transformed CSLEs and proved (making use of the corresponding prime SLEs solved under the DBCs) that the $\mathrm{R}\mathrm{R}\mathrm{Z}\text{T}\mathrm{s}$ of the eigenfunction do satisfy the DBCs unless the ExpDiffs lie within the LC range for both original or transformed CSLEs. Since the RRZTs of this kind turn the centrifugal barriers into infinitely deep potential wells, the violation of the conventional SUSY rules in the latter case [57] may not seem very exciting.

In [1], we presented a more physical example of the breakdown of the conventional rules of the SUSY quantum mechanics for the scenario when the ExpDiff for the pole at the origin lies within the LC range only for the transformed RCSLE, and, as a result, both initial and transformed potentials have centrifugal barriers at the origin. It was shown that this anomalous behavior of the transformed eigenfunctions was the direct manifestation of the fact that the first RRZT brought the quantum-mechanical system into the LC region, while the sequential transformation kept the system in this region, making the conventional rules unapplicable.

Let us mention once again that the use of the DBCs allows one to select only the mutually orthogonal PFSs, which is sufficient for quantum-mechanical applications. However, as an apparent setback of our approach, we were unable to pinpoint the X₁-Jacobi polynomials with two negative indexes larger than −1 (which represent the polynomial components of mutually orthogonal non-principal Frobenius solutions). It is very possible that the eigenfunction of the prime SLE (173) are accompanied by a finite set of similar mutually orthogonal non-principal Frobenius solutions formed by the $\mathrm{R}\mathrm{D}\text{T}\mathrm{s}$ of the R-Jacobi polynomials.

By analogy with the quasi-orthogonality of the X₁-Jacobi polynomials with the complex weight (78), one can extend the orthogonality relation (187) to the complex field provided that the complex indexes $*{\mathsf{\lambda }}_{\mathrm{o};\pm }$ of the EOPs are restricted by the condition $Re*{\mathsf{\lambda }}_{\mathrm{o};\pm }>0$. Our conclusion concerning the quasi-orthogonality of the $\mathrm{R}\mathrm{D}\text{T}\mathrm{s}$ of the R-Jacobi polynomials with the complex weight (186) was inspired by the work of Chen and Srivastava [34], who extended the conventional orthogonality condition for both classical Jacobi and R-Jacobi polynomials to the complex field (see the integrals (4.3) and (4.9) in [34]). Indeed, by analogy with the derivation of the orthogonality relation (187) under the real field, its extension to the complex indexes $*{\mathsf{\lambda }}_{\mathrm{o};\pm }$ of the EOPs under the constraint $Re*{\mathsf{\lambda }}_{\mathrm{o};\pm }>0$ directly follows from the fact that the symplectic form

$$\begin{gathered} [{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+],{\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+]]:= \\ ({\mathsf{\eta }}^2-1)\mathrm{W}\{\{{\cancel{\mathsf{\psi }}}_{\mathrm{j}}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+],{\cancel{\mathsf{\psi }}}_{{\mathrm{j}}^{\prime }}[\mathsf{\eta };*{\stackrel{\rightharpoonup }{\mathsf{\lambda }}}_{\mathrm{o}}|+]\}\mathrm{f}\mathrm{o}\mathrm{r} \mathsf{\eta }>1 \end{gathered}$$

(222)

vanishes for any pair of solutions of the complexified p-SLE (163) solved under the DBCs (175) and (176), provided that the complex indexes $*{\mathsf{\lambda }}_{\mathrm{o};\pm }$ have positive real parts:

$$-\mathsf{\pi }/2<\mathit{arg}*{\mathsf{\lambda }}_{\mathrm{o};\pm }<+\mathsf{\pi }/2$$

(223)

and lie outside the real interval [1, ∞]. Representing (A37) as

$$b(\mathsf{\beta },\mathsf{\alpha })={\displaystyle \frac{|\mathsf{\alpha }{|}^2-|\mathsf{\beta }{|}^2+Im(\mathsf{\alpha }\mathsf{\beta }*)}{|\mathsf{\beta }-\mathsf{\alpha }{|}^2}}$$

(224)

we conclude that the third pole (34) of the CSLE (35) lies on the real axis iff

$$\mathit{arg}*{\mathsf{\lambda }}_{\mathrm{o};+}=\mathit{arg}*{\mathsf{\lambda }}_{\mathrm{o};-}$$

(225)

which gives

$$b(*{\mathsf{\lambda }}_{\mathrm{o};-},*{\mathsf{\lambda }}_{\mathrm{o};+})={\displaystyle \frac{|*{\mathsf{\lambda }}_{\mathrm{o};-}|+|*{\mathsf{\lambda }}_{\mathrm{o};+}|}{|*{\mathsf{\lambda }}_{\mathrm{o};-}|-|*{\mathsf{\lambda }}_{\mathrm{o};+}|}}$$

(226)

More specifically, the third pole (34) of the CSLE (35) for σ= + is located on the negative semi-axis if we choose

$$|*{\mathsf{\lambda }}_{\mathrm{o};+}|<|*{\mathsf{\lambda }}_{\mathrm{o};-}|$$

(227)

Note that the third pole always lies outside the real interval [−1,+1] for σ = −. We conclude that the infinite quasi-orthogonal sequences of the complex X₁-Jacobi polynomials exist for any pair of the complex indexes $*{\mathsf{\lambda }}_{\mathrm{o};+}$ with positive real parts. As for their finite counter-parts, they exist either if $\mathit{arg}*{\mathsf{\lambda }}_{\mathrm{o};+}\ne \mathit{arg}*{\mathsf{\lambda }}_{\mathrm{o};-}$ or, alternatively, if the absolute values of the complex indexes $*{\mathsf{\lambda }}_{\mathrm{o};\pm }$ are restricted by the constraint (227).

It is very possible that the quasi-orthogonality relations (187) also hold if the first index lies within the interval (−1,0], but, again, our approach failed to address this issue.

It remains unclear whether the discovered complex q-RSs have any physical meaning if one complexifies the corresponding Liouville potential in the specified fashion. Since the complex solutions in question are quasi-orthogonal (not orthogonal!), we doubt that they can represent the eigenfunctions of any properly formulated SLP, and this was the main reason for removing this part of the discussion from the main body.

9. Conclusions

As demonstrated above, there are several rationales for a separate examination of the X₁-Jacobi DPSs and the corresponding infinite and finite subsets of the EOPs, before proceeding to the general case of the X_m-OPSs [45,46] and the finite EOP subsequences of the X_m-Jacobi DPSs [11]. To fully understand the significance of the presented results, the reader should keep in mind that there are the four X_m-Jacobi DPSs of series D, W, J1 and J2 for m > 1 which, as proven here, merge into the two DPSs of series J and D for m = 1. In particular, the X_m-DPSs of series J1 and J2 merge into the single X-DPS for m = 1, which contains both X₁-Jacobi OPS and the finite EOP sequences composed of the $\mathrm{RD}\text{T}\mathrm{s}$ of the R-Jacobi polynomials. As a result, the term ‘X₁-Jacobi polynomials’, put forward by Yadav et al. [37] as the synonym for the X₁-Jacobi DPS in our terms, has an unambiguous meaning, in contrast to the more equivocal epithet ‘X_m-Jacobi polynomials’ [85].

The revealed double-step shape-invariance of the renowned [8,9] Liouville potentials quantized by infinite [56] and finite [11,27] orthogonal sequences of X₁-Jacobi polynomials constitutes another important reason for treating the X₁-Jacobi DPSs separately from the general case m >1.

Since both the X₁-Jacobi DPSs are formed by Heun polynomials, they can be used as ‘guinea pigs’ in the theory of polynomial solutions of the Heun equation. The fact that the transformed CSLE has only one extraneous pole also made it easier to study its location and to unambiguously avoid the cases when it lies inside the quantization interval. In particular, one can readily prove the quasi-orthogonality of the complex polynomials obtained by the complexification of the X_m-Jacobi OPSs and the finite EOP subsets of X_m-Jacobi DPSs; however, the elimination of the cases when at least one of the poles lies inside the quantization interval represents a more challenging problem.