Generating Functions for Some Families of the Generalized Al-Salam-Carlitz $q$-Polynomials

In this paper, by making use of the familiar $q$-difference operators $D_q$ and $D_{q^{-1}}$, we first introduce two homogeneous $q$-difference operators $\mathbb{T}({\bf a},{\bf b},cD_q)$ and $\mathbb{E}({\bf a},{\bf b}, cD_{q^{-1}})$, which turn out to be suitable for dealing with the families of the generalized Al-Salam-Carlitz $q$-polynomials $\phi_n^{({\bf a},{\bf b})}(x,y|q)$ and $\psi_n^{({\bf a},{\bf b})}(x,y|q)$. We then apply each of these two homogeneous $q$-difference operators in order to derive generating functions, Rogers type formulas, the extended Rogers type formulas and the Srivastava-Agarwal type linear as well as bilinear generating functions involving each of these families of the generalized Al-Salam-Carlitz $q$-polynomials. We also show how the various results presented here are related to those in many earlier works on the topics which we study in this paper.


INTRODUCTION, DEFINITIONS AND PRELIMINARIES
The quantum (or q-) polynomials constitute a very interesting set of special functions and orthogonal polynomials. Their generating functions appear in several branches of mathematics and physics (see, for details, [1,2,3,4,5]), such as (for example) continued fractions, Eulerian series, theta functions, elliptic functions, quantum groups and algebras, discrete mathematics (including combinatorics and graph theory), coding theory, and so on.
In the year 1997, Chen and Liu [10] developed a method of deriving basic (or q-) hypergeometric identities by parameter augmentation, which may be viewed as being analogous to the method used rather extensively in the theory of ordinary hypergeometric functions and hypergeometric generating functions (see, for details, [5]). Subsequent investigations along the lines developed in [10] can be found in [11], [12], [13], [14], [15] and [16]. The main object of this paper is to investigate two families of the generalized Al-Salam-Carlitz q-polynomials φ (a,b) n (x, y|q) and ψ (a,b) n (x, y|q) by first representing them by the homogeneous q-difference operators T(a, b, cD q ) and E(a, b, cD q −1 ), which we have introduced here. We then derive a number of q-identities such as (among other results) generating functions, Rogers type formulas, two kind of the extended Rogers type formulas, and Srivastava-Agarwal type generating functions for each of the generalized Al-Salam-Carlitz q-polynomials φ (a,b) n (x, y|q) and ψ (a,b) n (x, y|q).
Here, in this paper, we adopt the common conventions and notations on q-series and q-hypergeometric functions. For the convenience of the reader, we provide a summary of the mathematical notations and definitions, basic properties and other relations to be used in the sequel. We refer, for details, to the general references (see [17,2,18]) for the definitions and notations. Throughout this paper, we assume that |q| < 1.
Also, for m large, we have The q-numbers and the q-factorials are defined as follows: The q-binomial coefficient is defined as follows (see, for example, [2]): The basic (or q-) hypergeometric function of the variable z and with r numerator and s denominator parameters is defined as follows (see, for details, the monographs by Slater [ [20] and [17]): where q = 0 when r > s + 1. We also note that Here, in our present investigation, we are mainly concerned with the Cauchy polynomials p n (x, y) as given below (see [21] and [2]): together with the following Srivastava-Agarwal type generating function (see also [29]): For λ = 0 in (1.5), we get the following simpler generating function [21]: The generating function (1.6) is also the homogeneous version of the Cauchy identity or the following q-binomial theorem (see, for example, [2], [19] and [18]): Upon further setting a = 0, this last relation (1.7) becomes Euler's identity (see, for example, [2]): and its inverse relation given below [2]: The Jackson's q-difference or q-derivative operators D q and D q −1 are defined as follows (see, for example, [22,23,2]): (1.10) Evidently, in the limit when q → 1−, we have provided that the derivative f ′ (x) exists.
Suppose that D q acts on the variable a. Then we have the q-identities asserted by Lemma 1 below.
Lemma 1. Each of the following q-identities holds true for the q-derivative operator D q acting on the variable a: (1.12) (1.14) The Leibniz rules for the q-derivative operators D q and D q −1 are given by (see, for example, [10] and [12]) where D 0 q and D 0 q −1 are understood to be the identity operators.
Lemma 2. Suppose that q-difference operator D q acts on the variable a. Then Proof. Suppose that the operator D q acts upon the variable a. In light of (1.15), we then find that Similarly, by using the relation (1.16), we have We now state and prove the q-difference formulas asserted by Theorem 1 below.
Theorem 1. Suppose that the q-difference operators D q and D q −1 act upon the variable a. Then Proof. Suppose first that the q-difference operator D q acts upon the variable a. Then, in light of (1.15), and by using the relations (1.17) and (1.11), it is easily seen that Similarly, by using (1.16), we find for the q-difference operator D q −1 acting on the variable a that q −n , s/ω, q/(at); q k q, q/(aω); q k q k , This paper is organized as follows. In Section 2, we introduce two homogeneous q-difference operators T(a, b, cD q ) and E(a, b, cD q −1 ). In addition, we define two families of the generalized Al-Salam-Carlitz q-polynomials φ (a,b) n (x, y|q) and ψ (a,b) n (x, y|q) and represent each of the families in terms of the homogeneous q-difference operators T(a, b, cD q ) and E(a, b, cD q −1 ). We also derive generating functions for these families of the generalized Al-Salam-Carlitz q-polynomials. In Section 3, we first give the Rogers type formulas and the extended Rogers type formulas. The Srivastava-Agarwal type generating functions involving the generalized Al-Salam-Carlitz q-polynomials are derived in Section 4. Finally, in our last section (Section 5), we present the concluding remarks and observations concerning our present investigation.
We now derive the identities (2.3) and (2.4) below, which will be used later in order to derive the generating functions, the Rogers type formulas, the extended Rogers type formulas and the Srivastava-Agarwal type generating functions involving the families of the generalized Al-Salam-Carlitz q-polynomials.
Definition. In terms of the q-binomial coefficient, the families of the generalized Al-Salam-Carlitz qpolynomials φ Proposition. Suppose that the operators D q and D q −1 act on the variable y. Then In our proof of Theorem 3, the following easily derivable Lemma will be needed.
Lemma 4. Suppose that the operators D q and D q −1 act on the variable a. Then Proof of Theorem 3. We suppose that the q-difference operator D q acts upon the variable y. In light of the formulas in (2.9), and by applying (2.12), it is readily seen that (2.14) Similarly, we have The proof of Theorem 3 can now be completed by making use of the relation (2.13).

THE ROGERS TYPE FORMULAS AND THE EXTENDED ROGERS TYPE FORMULAS
In this section, we use the assertions in (2.9) to derive several q-identities such as the Rogers type formulas and the extended Rogers type formulas for the families of the generalized Al-Salam-Carlitz q-polynomials φ  (a 1 , a 2 , · · · , a r+1 ; q) n (q, b 1 , b 2 , · · · , b r ; q) n (xt) n Theorem 5. Rogers type formula for ψ (a,b) n (x, y|q) The following Rogers type formula holds true for ψ (a,b) n (x, y|q): In order to prove Theorems 4 and 5, we need Lemma 5 below.
Lemma 5. It is asserted that Proof of Theorems 4 and 5. We suppose that the operator D q acts upon the variable y. Then, in view of the formulas in (2.9), we have The proof of the assertion (3.1) of Theorem 4 can now be completed by using the relation (3.3) in (3.5).
Similarly, we observe that which evidently completes the proof of the assertion (3.2) of Theorem 5.
We next derive another Rogers type formula for the family of the generalized Al-Salam-Carlitz qpolynomials ψ (a,b) n (x, y|q) as follows.
Theorem 6. Another Rogers type formula for ψ Proof. We suppose that the q-difference operator D q acts upon the variable y. We then obtain The proof of the assertion (3.7) of Theorem 6 can now be completed by applying the formula (2.4) with s = 0 and ω = s in (3.8).
Another extended Rogers type formula for the family of the generalized Al-Salam-Carlitz q-polynomials Ψ (a,b) n (x, y|q) is given by Theorem 7 below. Theorem 7. Another extended Rogers type formula for ψ (a,b) n (x, y|q) It is asserted that The proof of the assertion (4.7) of Theorem 9 is now completed by making use of the relation (2.3) in (4.8).
Theorem 10. Srivastava-Agarwal type bilinear generating function for ψ (a,b) n (x, y|q) The following Srivastava-Agarwal type bilinear generating function holds true for the family of the generalized Al-Salam-Carlitz q-polynomials ψ (a,b) n (x, y|q): (4.9) Proof. We suppose that the q-difference operator D q acts upon the variable y. We then obtain The proof of the assertion (4.9) of Theorem 10 can now be completed by making use of the relation (2.4) in (4.10).

CONCLUDING REMARKS AND OBSERVATIONS
Our present investigation is motivated essentially by several recent studies of generating functions and other results for various families of basic (or q-) polynomials stemming many from the works by Hahn (see, for example, [24], [25] and [26]; see also Al-Salam and Carlitz [27], Srivastava and Agarwal [28], Cao and Srivastava [29], and other researchers cited herein).
In terms of the familiar q-difference operators D q and D q −1 , we have first introduced two homogeneous q-difference operators T(a, b, cD q ) and E(a, b, cD q −1 ), which turn out to be suitable for dealing with the generalized Al-Salam-Carlitz q-polynomials φ (a,b) n (x, y|q) and ψ (a,b) n (x, y|q). We have then applied each of these two homogeneous q-difference operators in order to derive generating functions, Rogers type formulas, the extended Rogers type formulas and the Srivastava-Agarwal type linear and bilinear generating functions for each of these families of the generalized Al-Salam-Carlitz q-polynomials.
The various results, which we have presented in this paper, together with the citations of many related earlier works are believed to motivate and encourage interesting further researches on the topics of study here.
In conclusion, it should be remarked that, in a recently-published survey-cum-expository article, Srivastava [30] presented an expository overview of the classical q-analysis versus the so-called (p, q)analysis with an obviously redundant additional parameter p (see, for details, [30, p. 340]).