Some Generating Functions for q-Polynomials

Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain q-analogues for some of their generating functions. Our q-generating functions are given in terms of the basic hypergeometric series 4ϕ5, 5ϕ5, 4ϕ3, 3ϕ2, 2ϕ1, and q-Pochhammer symbols. Starting with our q-generating functions, we are also able to find some new classical generating functions for the Pasternack and Bateman polynomials.

He also obtained the generating functions ∞ n=0 B n (z) t n = 1 Lemma 1.1. Let t ∈ C. Then the following relation holds: The Bateman polynomial B n was generalized by Pasternack in [6]. He defines the Pasternack polynomial B m n as B m n (z) = 3 F 2 −n, n + 1, z+m+1 for m ∈ C\{−1}. The Pasternack polynomials reduce to the Bateman polynomials when m = 0. Further information regarding such polynomials and their connection with (classical) orthogonal polynomials can be found in [5]. Indeed, we can write the Pasternack polynomials in terms of the continuous Hahn polynomials [4, (9.4.1)] as follows [5, p. 893]: We also consider the Sylvester polynomials, defined as (see [8, p. 185]) Notice that we also can write the Sylvester polynomials in terms of (classical) orthogonal polynomials [4] ϕ Here L (α) n and C n represent the Laguerre and Charlier polynomials, respectively. It is also known that the Sylvester polynomials satisfy the generating functions The Cesàro polynomials are defined as [8, p. 449] Observe that this family can be written in terms of Jacobi polynomials [8, p. 449] as g (s) n (z) = P (s+1,−s−n−1) n (2z − 1). Furthermore, they satisfy the generating functions where ℓ = 0, 1, 2, .... The aim of this paper is to obtain q-analogues of all these families of polynomials as well as q-analogues of the generating functions stated above. The structure of this paper is as follows. In Section 2 we give some preliminaries on q-calculus and we define some q-analogues of Bateman-Z , Bateman, Sylvester, Pasternack, and Cesàro polynomials. In Section 3 we state and derive q-analogues of the given generating functions associated with the q-Bateman-Z , q-Bateman, q-Sylvester, q-Pasternack, and q-Cesàro polynomials.
Let q ∈ C, 0 < |q| < 1. A q-analogue of the hypergeometric series p F r is the basic hypergeometric series [3] where q = 0 when r > s + 1, and the (b i ) are such that the denominator never vanishes. We also need to define some other q-analogues, such as the q-analogue of a number [a] q , factorial [a] q !, and the Pochhammer symbol (rising factorial), (a) n . These q-analogues are given as follows: and the q-binomial coefficient Taking into account the previous definitions and results, we define q analogues of the above introduced polynomials. We define Z n (z; q) = 2 φ 2 q −n , q n+1 q, q ; q, q n z , and the q-Bateman polynomial as the q-Sylvester polynomial as the q-Cesàro polynomial as Remark 2.1. Note that B 0 n (z; q) = B n (z; q).
Lemma 2.2. The q-Cesàro polynomial can be written as

The Generating Functions
Theorem 3.1. For any t ∈ C small enough, the q-Bateman-Z and the q-Bateman polynomials satisfy the following generating functions: ; q, qt .
By using the identity Next, we rearrange the double summation and set n → n + k, obtaining By using the identity (a; q) 2n = (a Hence the identity follows. In order to prove identity (3.3): Taking into account that the above expression vanishes at k = 0 we set k → k + 1, rearrange the double sum, and set n → n + k, yielding Here, again, the series vanishes at n = 0 so we set n → n + 1. Applying some basic identities of q-Pochhammer symbols, we obtain (q 3+2k ; q) n (q; q) n t n .
Applying again (2.1) and simplifying one has Therefore the identity follows. Let us prove the generating function (3.3). We have As in the previous identities, we rearrange the double sums and set n → n + k obtaining Hence the identity follows.
In fact, we obtain a new generating function for the Pasternack and Bateman polynomials by taking the same limit q ↑ 1.
Corollary 3.3. The following identities hold: