1. Introduction
The Al-Salam–Carlitz I and II orthogonal polynomials of degree
n, usually denoted in the literature as
and
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]. Here,
stands for a fixed parameter, being the polynomials written in the variable
x. There is a straightforward relationship between
and
(see [
2] (Chapter VI, §10, pp. 195–198))
and they are known to be positive definite orthogonal polynomial sequences for
, and
respectively. Here, and throughout the paper, we assume the parameter
q is such that
, which implies that these 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 [
3,
4]). In fact, as we show later on, they can be explicitly 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 on the Al-Salam–Carlitz I orthogonal polynomials
. The results obtained can be also stated for the Al-Salam–Carlitz II orthogonal polynomials by replacing the parameter
q by
, so we omit explicit details concerning this second family of orthogonal polynomials.
The Al-Salam–Carlitz I polynomials are orthogonal on the interval
, 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 [
5], 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 ([
6] (Section 18.2)), with birth rate
and death rate
, and for
they become the well known Rogers-Szegő polynomials, of deep implications in the study of the celebrated Askey–Wilson integral (see, for example [
7,
8]).
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 [
9] 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 [
10,
11,
12], and also [
13] 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 Sobolev-type 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 [
14] for only one
q-derivative). In the present study, we consider an arbitrary number
,
of
q-derivatives in the discrete part of the inner product. For an interesting related work to this paper, see, for example, the preprint [
15], 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 [
16], and also in the present paper. It is also worth mentioning the nice variation considering special non-uniform lattices (snul), instead of uniform or
q-lattices, studied in the recent work [
17].
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 reveals some small 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 [
15]), and that we have right now under study. To be more precise, this paper deals with the sequence of monic
q-polynomials
, orthogonal with respect to the Sobolev-type inner product
where
is the orthogonality measure associated to the Al-Salam–Carlitz I orthogonal polynomials,
,
and
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
and
, 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
as Al-Salam–Carlitz I-Sobolev type orthogonal polynomials of higher order, and for the sake of brevity, in what follows we just write
. We provide here two explicit representations for
, one as a linear combination of two consecutive Al-Salam–Carlitz I orthogonal polynomials
and
, and the other one as a
q-hypergeometric series, which was unknown so far. This basic hypergeometric character is always
, 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
, 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
of precise degree
, in terms of the previous two consecutive polynomials of the same sequence
and
, and up to four different versions of the linear second order
q-difference equation satisfied by
.
In the work, we provide four different versions of the second order q-difference equations satisfied by the family of orthogonal polynomials under consideration. Also, two different representations of such polynomials are determined: one as a linear combination of standard Al-Salam–Carlitz I orthogonal polynomials, and a second one as series. Two versions of structure relations are obtained, in contrast to standard results, giving rise to four second order q-difference equations satisfied by the elements of this family. The previous results, which clarify the enriching structure of such polynomials, are followed by novel results using a non-standard technique to achieve a three-term recurrence formula, and leading to the appearance of the polynomials under consideration as the numerators and denominators of the convergents of certain J-fractions.
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
, as well as the two different versions of the aforementioned three term recurrence formula with rational coefficients that
satisfies. In
Section 5, combining the connection formula for
, 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.
3. Connection Formulas and Hypergeometric Representation
In this section we define the Al-Salam–Carlitz I-Sobolev type polynomials of higher order
, 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
,
, and
,
.
In a first approach, we express
in terms of the Al-Salam–Carlitz I polynomials
, the kernel polynomials and their corresponding derivatives. Moreover, we obtain a representation of the proposed polynomials as hypergeometric functions. Let us depart from the Fourier expansion
In view of (
15), and considering the orthogonality properties for
, for
, the coefficients in the previous expansion are given by
Thus
After some manipulations, we obtain a linear system
with two unknowns, namely
and
, where
and
Cramer’s rule yields
where
Hence, we obtain a first connection formula, namely
where
and
The previous connection formula is obtained after the application of Lemma 1, which yields
together with
Therefore, from (
16) we get
leading to (
17). 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
and
From (
17) and (
18) we deduce
and
where
Finally, we focus our attention on the representation of
as hypergeometric functions. A similar analysis to that carried out in [
14] (Theorem 2), yields the following result
Proposition 3 (Hypergeometric character).
For , the Al-Salam–Carlitz I-Sobolev type polynomials of higher order , have the following hypergeometric representation:where and Proof. For
, a trivial verification shows that (
21) yields
. For
, combining (
8) with (
17) and the relations
where, in order to improve the compactness and readability of the expressions involved, we define
and
yields
More precisely, (
8) together with (
17) yield
Taking into account (
23), the previous expression turns into
and (
22) now leads to
Finally, a rearrangement of the terms in the sums of the previous expression, leads to (
24).
On the other hand, after some straightforward calculations we get
Therefore
which coincides with (
21). This completes the proof. □
Remark 2. Notice that one recovers (8) from (21) after and . One also recovers [14] ((23), p. 13) for , and in (21). 4. Ladder Operators and a Three Term Recurrence Formula
In this section we find several structure relations associated to . 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 () whether in the other two, a q-contraction operator determines the q-derivative (). The ladder (creation and annihilation) operators are obtained in Proposition 4, as well as the three-term recurrence relations of Theorem 2, satisfied by . 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 relations stated in Theorem 1 lean on the following result.
Lemma 2. Let be the sequence of Al-Salam–Carlitz I-Sobolev type polynomials of degree n. Then, following statements hold, for ,andwhere for and otherwise. Moreover,andHere, denote the Kronecker delta function. Proof. It is a direct consequence of the connection Formulas (
17)–(
20), the three-term recurrence relation (
9) satisfied by
, and the structure relation (
10). To be more precise, applying the
q-derivative operator
to (
17) for
, together with the property (
5) yields
Thus, multiplying the above expression by
for
, and next combining (
10) with (
11) and (
9), we deduce (
25). Finally, shifting the index in (
25) as
and using the recurrence relation (
9) we get (
26). This completes the proof. □
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.
Theorem 1. The Al-Salam–Carlitz I-Sobolev type polynomials of high order satisfy the following structure relations for ,andwhere and Proof. The result follows in a straightforward way from (
19) and (
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 [
16].
Proposition 4. Let 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 and , respectively, are defined bywhich verifywhere . We complete this section with a straightforward application of the above ladder operators
and
. We use these operators to obtain two versions (one for
and other for
) of certain three term recurrence formula with rational coefficients which provides
in terms of the former two consecutive polynomials
and
. We recall the importance of such recurrence formula, which allows to give further properties of the family of orthogonal polynomials, as described in classical references such as [
2]. The proof of the next result can be followed from the aforementioned technique, which has been recently generalized in [
16] (p. 8). However, we have decided to include it below for the sake of completeness.
Theorem 2. The Al-Salam–Carlitz I-Sobolev type polynomials of high order satisfy the following three-term recurrence relations for ,whereand Proof. Shifting the index in (
28) as
, yields
Next, multiplying the above expression by
, and multiplying (
27) by
, adding and simplifying the resulting equations, we obtain (
33). This completes the proof. □
Remark 3. Notice that (33) becomes (9) when .