On second order q-difference equations satisfied by Al-Salam-Carlitz I-Sobolev type polynomials of higher order

This contribution deals with the sequence $\{\mathbb{U}_{n}^{(a)}(x;q,j)\}_{n\geq 0}$ of monic polynomials, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam--Carlitz I orthogonal polynomials, and involving an arbitrary number of $q$-derivatives on the two boundaries of the corresponding orthogonality interval. We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order $q$-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by $\mathbb{U}_{n}^{(a)}(x;q,j)$, which paves the way to establish an appealing generalization of the so-called $J$-fractions to the framework of Sobolev-type orthogonality.


Introduction
The Al-Salam-Carlitz I and II orthogonal polynomials, usually denoted in the literature as U n (x; q) respectively, are two systems of one parameter q-hypergeometric polynomials introduced in 1965 by W. A. Al-Salam and L. Carlitz, in their seminal work [1]. There is a straightforward relationship between U  n (x; q) (see [9,Ch. VI,§10,) U (a) n (x; q −1 ) = V (a) n (x; q), and they are known to be positive definite orthogonal polynomial sequences for a < 0, and a > 0 respectively. Here, and throughout the paper, we assume 0 < q < 1, which implies that this two families belong to the class of orthogonal polynomial solutions of certain second order q-difference equations, known in the literature as the Hahn class (see [16], [20]). In fact, as we show later on, they can be explicity given in terms of basic hypergeometric series. Given the close relation between these two families, and for the sake of clarity, in this paper we will focus only on the Al-Salam-Carlitz I orthogonal polynomials {U (a) n (x; q)} n≥0 . They are orthogonal on the interval [a, 1], with a quite simple q-lattice, which makes them suitable for the study to be carried out hereafter, and also they are of interest in their own right. For example, they are known to be proportional to the eigenfunctions of certain quantum mechanical q-harmonic oscillators. In [4], it is clearly shown that many properties of this q-oscillators can be obtained from the properties of the Al-Salam-Carlitz I orthogonal polynomials. They are also known to be birth and death process polynomials ( [18,Sec. 18.2]), with birth rate aq n and death rate 1−q n , and for a = 0 they become the well known Rogers-Szegő polynomials, of deep implications in the study of the celebrated Askey-Wilson integral (see, for example [3], [19]).
On the other hand, in the last decades, the so called Sobolev orthogonal polynomials have attracted the attention of many researchers. Firstly, this name was given to those families of polynomials orthogonal with respect to inner products involving positive Borel measures supported on infinite subsets of the real line, and also involving regular derivatives. When these derivatives appear only on function evaluations on a finite discrete set, the corresponding families are called Sobolev-type or discrete Sobolev orthogonal polynomial sequences. For a recent and comprehensive survey on the subject, see [22] and the references therein. In the last decade of the past century, H. Bavinck introduced the study of inner products involving differences (instead of regular derivatives) in uniform lattices on the real line (see [5], [6], [7], and also [17] for recent results on this topic). By analogy with the continuous case, these are also called Sobolev-type or discrete Sobolev inner products. In contrast, they are defined on uniform lattices. As a generalization of this last matter, here we focus on a particular Sobolevtype inner product defined on a q-lattice, instead of on a uniform lattice. This has already been considered in other works (see, for example in [10] for only one q-derivative). In the present study, we consider an arbitrary number j ∈ N , j ≥ 1 of q-derivatives in the discrete part of the inner product. For an interesting related work to this paper, see for example the preprint [13], which appeared just a few days ago while we were giving the finishing touches to the present manuscript. There, the authors generalize the action of an arbitrary number of q-derivatives for general orthogonality measures, using the same techniques as for example in [14], and also in the present paper. It worths mentioning as well the nice variation considering special non-uniform lattices (snul), instead of uniform or q-lattices, studied in the recent work [24].
Having said all that, and to the best of our knowledge, an arbitrary number of q-derivatives acting at the same time on the two boundaries of a bounded orthogonality interval, has never been previously considered in the literature, and the present work is intended to be a first step in this direction. This this seems to arise some differences of the corresponding polynomial sequences, for example related with the parity of the polynomials, with respect to what happens considering only one mass point (as in [13]), and that we have right now under study. This last interesting work appeared just a few days ago, while we were giving the finishing touches to the present manuscript. There, the authors generalize the action of an arbitrary number of q-derivatives for general orthogonality measures, using the same techniques as for example in [14], and also in the present paper.
To be more precise, this paper deals with the sequence of monic q-polynomials {U (a) n (x; q, j)} n≥0 , orthogonal with respect to the Sobolev-type inner product where (qx, a −1 qx; q) ∞ d q x is the orthogonality measure associated to the Al-Salam-Carlitz I orthogonal polynomials, a < 0, λ, µ ∈ R + and (D q f ) denotes the q-derivative operator, as defined below in (2). It is worth noting that the above inner product involves an arbitrary number of q-derivatives on function evaluations on the discrete points x = a and x = 1, exclusively. We observe such points conform the boundary of the orthogonality interval of the Al-Salam-Carlitz I orthogonal polynomials. Thus, as an extension of the language used in literature, throughout this manuscript we will refer to U (a) n (x; q, j) as Al-Salam-Carlitz I-Sobolev type orthogonal polynomials of higher order, and for the sake of brevity, in what follows we just write U (a) n (x; q). We provide here two explicit representations for U n (x; q) and U (a) n−1 (x; q), and the other one as a 3 φ 2 q-hypergeometric series, which was unknown so far. This basic hypergeometric character is always 3 φ 2 , with independence of the number j of q-derivatives considered in (1). Next, we obtain two different versions of the structure relation satisfied by the Sobolev-type q-orthogonal polynomials in U (a) n (x; q), and next we use them to obtain closed expressions for the corresponding ladder (creation and annihilation) q-difference operators. As an application of these ladder q-difference operators, we obtain a three-term recurrence formula with rational coefficients, which allows us to find every polynomial U  The manuscript is organized as follows. In Section 2, we recall some basic definitions and notations of the q-calculus theory, as well as the basic properties of the Al-Salam-Carlitz I polynomials. In Section 3, we obtain some connection formulas and the basic hypergeometric representation of the Al-Salam-Carlitz I-Sobolev type orthogonal polynomials of higher order. Section 4 is focused on two structure relations for the sequence {U n (x; q)} n≥0 , as well as the two different versions of the aforementioned three term recurrence formula with rational coefficients that U (a) n (x; q) satisfies. In Section 5, combining the connection formula for U (a) n (x; q), and the structure relations obtained in the preceeding Sections, we provide the q-difference ladder operators and four versions of the second linear q-difference equation that the Al-Salam-Carlitz I-Sobolev type polynomials of higher order satisfy. The work ends with two brief sections on further results. The first one describes results relating Al-Salam-Carlitz I-Sobolev type polynomials with Jacobi fractions, and the second illustrates the form of such polynomials together with some important remarks. A final section on conclusions and future research problems is also included.

Definitions and notations
This first part of the section is twofold. A first subsection provides the main tools used in the framework of q-calculus, in order to make our exposition be self-contained. Afterwards, we describe known facts on Al-Salam-Carlitz I polynomials.
The following definitions, also in the framework of the q-calculus, can be found in [20]. The basic hypergeometric, or q-hypergeometric series r φ s , is defined as follows. Let {a i } r i=1 and {b j } s i=1 be complex numbers such that b j = q −n for n ∈ N. For every j = 1, 2, . . . , s one writes The q-binomial coefficient is given by where n denotes a nonnegative integer. Concerning the q-analog of the derivative operator, we have the q-derivative, or the Euler-Jackson q-difference operator where D 0 Moreover, one has the following properties and the following interesting property which can be found in [21, p. 104] This q-derivative operator leads to define a q-analogue of Leibniz' rule Of special interest is the way in which the integral form of the inner product (1) is defined, corresponding to the so called Jackson q-integral, given by The following definition will also be needed throughout the paper. The Jackson-Hahn-Cigler q-subtraction is given by (see, for example [12,Def. 6], and the references given there) Finally, we recall here the q-Taylor formula (see [25,Th. 6.3]), with the Cauchy remainder term, which is defined by

Al-Salam-Carlitz I orthogonal polynomials
After the above q-calculus introduction, we continue by giving several aspects and properties of the Al-Salam-Carlitz I polynomials {U (a) n (x; q)} n≥0 . All the elements presented in the remaining of this section can be found in [9, Ch. VI, §10], [18,Sec. 18.2], [20,Sec. 14.24], and [15], among other references. Such polynomials are orthogonal with respect to the inner product on P, the linear space of polynomials with real coefficients and jumps −aq k (q/a; q) ∞ (q, aq; q) k at the points x = aq k , k = 0, 1, 2, . . . .
It can be easily checked that, when a = −1 the above inner product becomes the inner product associated to the discrete q-Hermite orthogonal polynomials. The Al-Salam-Carlitz I polynomials can be explicitly given by satisfying the orthogonality relation n (x; q)} n≥0 be the sequence of Al-Salam-Carlitz I polynomials of degree n. The following statements hold: 1. The recurrence relation [20] with initial conditions U Here, β n = (a + 1) q n and γ n = −aq n−1 (1 − q n ).
5. Second-order q-difference equation [11] σ n (x; q)} n≥0 be the sequence of Al-Salam-Carlitz I polynomials. If we denote the n-th reproducing kernel by Then, for all n ∈ N, it holds that Concerning the partial q-derivatives of K n,q (x, y), we use the following notation Next, we provide a technical result that will be useful later on.
n (x; q)} n≥0 be the sequence of Al-Salam-Carlitz I polynomials of degree n. Then following statements hold, for all n ∈ N, where Proof. Applying the j-th q-derivative to (12) with respect to y yields Then, using the q-analogue of Leibniz' rule (7) and Next, it is easy to check that and therefore we have Finally, combining all the above with (14) we obtain (13). This completes the proof.

Connection formulas and hypergeometric representation
In this section we define the Al-Salam-Carlitz I-Sobolev type polynomials of higher order {U n (x; q)} n≥0 , and describe different relations which relate them to the Al-Salam-Carlitz I polynomials. These links will be useful in the sequel. Al-Salam-Carlitz I-Sobolev type polynomials are defined to be orthogonal with respect to Sobolev-type inner product where a < 0, λ, µ ∈ R + , and j ∈ N , j ≥ 1.
In a first approach, we express {U Let us depart from the Fourier expansion k (x; q).
In view of (15), and considering the orthogonality properties for U (a) n (x; q), for 0 ≤ k ≤ n − 1, the coefficients in the previous expansion are given by n−1,q (x, 1).
After some manipulations, we obtain the following linear system AX = b with two unknowns, namely D j q U , and X = (D j q U (a) n (a; q), D j q U (a) n (1; q)) T , b = (D j q U (a) n (a; q), D j q U (a) n (1; q)) T . Cramer's rule yields where n (a; q) λK (j,j) Finally, from proposition 1, we obtain a first connexion formula, namely where j,n (a)A n (x, 1), and D 1,n (x) = −λ∆ (1) j,n (a)B n (x, a) − µ∆ (2) j,n (a)B n (x, 1). At this point, we provide another relation between the two families of polynomials, which will be applied in theorem 1. More precisely, from (17) and the recurrence relation (9) we have that where From (17)- (18) we deduce and U (a) where B n (x) = C 1,n (x) C 2,n (x) D 1,n (x) D 2,n (x) .
Finally, we focus our attention on the representation of U  n (x; q)} n≥0 , have the following hypergeometric representation: where ψ n (x) = ((1 − q)ϑ n (x) + 1) −1 and Proof. Combining (8) with (17) and the relations On the other hand, after some straightforward calculations we get which coincides with (21). This completes the proof.

Ladder operators and a three term recurrence formula
In this section we find several structure relations associated to {U (a) n (x; q)} n≥0 . It is worth mentioning that such relations can be grouped in two depending on nature of the action of the q−derivative involved in the relation (see theorem 1). In two of them, such q−derivative is constructed by means of a q−dilation operator (ℓ = −1) whether in the other two, a q−contraction operator determines the q−derivative (ℓ = 1). The ladder (creation and annihilation) operators are obtained in proposition 4, as well as the three-term recurrence relations of theorem 2, satisfied by {U (a) n (x; q)} n≥0 . These two results are also stated in terms of the duality provided by the choice of a q−dilating or q−contracting derivation.
The structure relation stated in theorem 1 leans on the following result.
As a direct consequence of the previous result, we next obtain the following structure relations for the Al-Salam-Carlitz I-Sobolev type polynomials of higher order. n (x; q)} n≥0 satisfy the following structure relations for ℓ = −1, 1, and where Θ ℓ,n (x) = σ ℓ (x) det(B n (x)) and Proof. The result follows in a straightforward way from (19)- (20), together with the application of the previous lemma 2.
In the next result, we provide ladder operators associated to the Al-Salam-Carlitz I-Sobolev type polynomials. Its proof is quite involved, leaning on the use of theorem 1, and following the same technique as in [14].
n (x; q)} n≥0 be the sequence of Al-Salam-Carlitz I-Sobolev type polynomials defined by (21), and let I be the identity operator. Then, the ladder (destruction and creation) operators a ℓ and a † ℓ , respectively, are defined by which verify where ℓ = −1, 1.

Holonomic second order q-difference equations
In this section, we find up to four different q-difference equations of second order that U (a) n (x; q) satisfies. It is worth noting that these q-difference equations are no longer classical, in the sense that all their polynomial coefficients depend on n, so we have different coefficients for every different degree n that we consider. When dealing with differential equations, these kind of equations are known in the literature as Holonomic second order differential equations (see, for example [14]), so by natural extension we refer to them as Second order linear q-difference equations.
It is also worth remarking the different nature of these four equations. The four of them are linear second order difference equations. However, in two of them (see proposition 5), a unique difference operator appears. We also point out the appearance of four regular singular points in one of these two equations, more precisely for the choice ℓ = −1. Such points are 1, a, q and aq, which appear to be intimately related to the problem. The two second difference equations (see proposition 6) an analogous disquisition can be made concerning the singular points, which are now the points 1, a, q −1 and aq −1 . Moreover, the two last equations involve the two q−difference operators previously mentioned, in contrast to the two first ones.

Jacobi Fractions and Al-Salam-Carlitz I-Sobolev type polynomials
In this section, we state some fresh results relating the elements in the family of orthogonal polynomials {U (a) n (x; q)} n≥0 and Jacobi fractions. Basic concepts and the main properties related to continued fractions, in particular with Jacobi fractions, and orthogonal polynomials can be found in [9], Chapter 3, for instance. In this section, we use our original result (30) to consider J−fractions in a more general sense than in the previous text. Given two sequences of polynomials with complex coefficients {a n (x)} n≥1 and {b n (x)} n≥0 , the J−fraction associated to the previous sequences is the formal expression For all n ≥ 0, the n-th convergent associated to the previous J−fraction is given by .
This can be proved directly from the definition of the differential operator D q applied on any monomial. Therefore, given j ≥ 1, one has that U (a) n (x; q, j) coincides with U (a) n (x; q) for all 0 ≤ n < j. The values of λ, µ are irrelevant for these first polynomials in the sequence. This phenomenon can also be observed in different points through the work. More precisely, the representation (16) of U  n (x; q) is made in terms of the quantities ∆ (i) j,n (a), for i = 1, 2. The definition of these two elements is made in terms of the determinant of a matrix with a null column for j > n. Therefore, formula (16) states the coincidence of Al-Salam-Carlitz I polynomials and the Sobolev type polynomials. This property is also directly observed at the Fourier coefficients a n,k .
We also remark that the choice of λ = µ = 0 provides U (a) n (x; q, j) = U (a) n (x; q) for every n. We observe this is the case for the polynomial U (a) 3 (x; q, 2), which is given by U (a) 3 (x; q, 2) = x 3 + (−aq 2 − aq − q 2 − a − q − 1)x 2 + (a 2 q 3 + a 2 q 2 + aq 3 + a 2 q + 2 aq 2 + q 3 + 2 aq + q 2 + a + q)x − a 3 q 3 − a 2 q 3 − a 2 q 2 − aq 3 − a 2 q − aq 2 − q 3 − aq, after evaluation at λ = µ = 0.  n (x; q, j)} n≥0 , orthogonal with respect to a Sobolev-type inner product associated to the Al-Salam-Carlitz I orthogonal polynomials. The studied inner product involves an arbitrary number of q-derivatives, evaluated on the two boundaries of the orthogonality interval of the Al-Salam-Carlitz I orthogonal polynomials. We gave two representations for U (a) n (x; q, j), one as a linear combination of two consecutive Al-Salam-Carlitz I orthogonal polynomials, and other as a 3 φ 2 series. We state not only as usual, but two different versions of structure relations, which lead to the corresponding ladder operators, which help us to find up to four different versions of the second order linear q-difference equation satisfied by U (a) n (x; q, j). Finally, as a truly original contribution to the literature, we obtained a three term recurrence formula with rational coefficients satisfied by U (a) n (x; q, j), which is the key point to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality. As problems to be addressed in a future contribution, we consider to analyze the effect of having two mass points, each one on a different side of the bounded orthogonality interval, in the parity of the corresponding Sobolev-type orthogonal sequence. We also whish to carry out an in-depth analysis on the zero behavior of these polynomials, as well as to study several of their asymptotic properties.