A General Family of $q$-Hypergeometric Polynomials and Associated Generating Functions

In this paper, we introduce a general family of $q$-hypergeometric polynomials and investigate several $q$-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of $q$-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized $q$-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various $q$-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called $(p,q)$-variations of the $q$-results, which we have investigated here, because the additional parameter $p$ is obviously redundant.

Recently, Cao [6, Eq. (4.7)] introduced the following two families of generalized Al-Salam-Carlitz polynomials: together with their generating functions given by (see [6,Eqs.(4.10) and (4.11)]) Remark 1.For (b, c, y) = (0, 0, 1) and For (a, b, c, y) = (1/a, 0, 0, 1), (1.12) and (1.13) reduce to φ respectively.More recently, Srivastava and Arjika [22] introduced two families φ of the generalized Al-Salam-Carlitz q-polynomials, which are defined by respectively, where On the other hand, Cao [6] introduced and studied the following family of q-polynomials: where together with their generating function given by Our present investigation is motivated essentially by the earlier works by Srivastava and Arjika [22] and Cao [6].Our aim here is to introduce and study some further extensions of the above-mentioned q-polynomials.Definition 1.In terms of q-binomial coefficient, a family of generalized q-hypergeometric polynomials Ψ (a,b) n (x, y, z|q) are defined by where, for convenience, The above-defined q-polynomials Ψ (a,b) n (x, y, z|q) include many one-variable q-hypergeometric r Φ s series as special or limit cases.Therefore, we choose just to call them generalized q-hypergeometric polynomials.

Generalized q-Hypergeometric Polynomials
In this section, we begin by introducing a homogeneous q-difference hypergeometric operator as follows.
Definition 2. The homogeneous q-difference hypergeometric operator is defined by 1+s−r . (2.1) We now derive the q-series identities (2.2), (2.3) and (2.4) below, which will be used later in order to derive the extended generating functions, the Rogers type formulas and the Srivastava-Agarwal type bilinear generating functions involving the generalized q-hypergeometric polynomials.
We now derive an extended generating function for the generalized q-hypergeometric polynomials Ψ (a,b) n (x, y|q) by using the operator representation (2.2).
Remark 3. Setting k = 0 in Theorem 1, we get the following generating function for the generalized q-hypergeometric polynomials: (2.8)

The Rogers Formula
In this section, we use the assertion (2.4) of Lemma 1 in order to derive several q-identities such as the Rogers type formula for the generalized q-hypergeometric polynomials Ψ (a,b) n (x, y|q).
Theorem 2 (The Rogers formula for Ψ (a,b) n (x, y, z|q)).For max t ω , |yω| < 1, the following Rogers type formula holds: Proof.In light of (2.2), we have Now, by using (2.4) as well as (1.2), this last relation takes the following form: which evidently completes the proof of Theorem 2.
4. The Srivastava-Agarwal Type Bilinear Generating Functions for the Generalized q-Hypergeometric Polynomials Ψ (a,b) n (x, y, z|q) In this section, by applying the following homogeneous q-difference hypergeometric operator [17]: we derive the Srivastava-Agarwal type generating functions for the generalized q-hypergeometric polynomials Ψ (a,b) n (x, y, z|q) defined by (1.22).We also deduce a bilinear generating function for the Al-Salam-Carlitz polynomials ψ (α) n (x|q) as an application of the Srivastava-Agarwal type generating functions.
We now state and prove the Srivastava-Agarwal type bilinear generating functions asserted by Theorem 3 below.

A Transformational Identity Involving Generating Functions for the Generalized q-Hypergeometric Polynomials
In this section, we derive the following transformational identity involving generating functions for the generalized q-hypergeometric polynomials.Once again, in our derivation, we apply the homogeneous q-difference operator (2.1).Theorem 4. Let A(n) and B(n) satisfy the following relationship: provided that each of the series in (5.1) and (5.2) is absolutely convergent.

Concluding Remarks and Observations
In our present investigation, we have introduced a general family of q-hypergeometric polynomials and we have derived several q-series identities such as an extended generating function and Srivastava-Agarwal type bilinear generating functions for this family of q-hypergeometric polynomials.We have presented a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here.We have also pointed out relevant connections of the various q-results, which we have investigated in this paper, with those in several related earlier works on this subject.

n
(x) defined in(1.11).Motivated by the above-cited work[6], Cao et al.[8] introduced the extensions φ ( y|q) of the Al-Salam-Carlitz polynomials, which are defined by y, z|q) defined in (1.22) are generalized and unified form of the Hahn polynomials and the Al-Salam-Carlitz polynomials.