Some New Generating Functions forq-Hahn Polynomials

Abstract

We obtain some new generating functions forq-Hahn polynomials and give their proofs based on the homogeneousq-difference operator.

Generalized q-Difference Equations for q-Hypergeometric Polynomials with Double q-Binomial Coefficients

“…Saad and Abdlhusein [13] utilized the Cauchy operator in proving some identities involving the homogeneous Rogers-Szegö polynomials. However, we found it to be difficult to continue to calculate and generalize the above-mentioned authors' results for general q-polynomials with more parameters (see, for example, [10,[12][13][14][15]).…”

“…The novelty of this paper is to search and find these generalized q-difference equations that are satisfied by some of the general q-hypergeometric polynomials, which we have investigated in this paper. The methods and techniques, which we have presented and used here, have produced potentially useful generalizations of the above-mentioned results (see, for details, [10,[12][13][14][15]). Derivations of various known or new particular cases of our results are indicated in Remark 1.…”

“…where τ(r, s) is defined as in (9). Zhou and Luo [12] obtained some new generating functions for the q-Hahn polynomials and their proofs are based upon the homogeneous q-difference operator. Saad and Abdlhusein [13] utilized the Cauchy operator in proving some identities involving the homogeneous Rogers-Szegö polynomials.…”

“…Remark 1. The general q-polynomials L(r,s) n (α, x, y, z, a, b, c) defined in (12) provide a generalized and unified form of the Hahn polynomials and the Al-Salam-Carlitz polynomials. Some of these special cases of the general q-polynomials L(r,s) n (α, x, y, z, a, b, c) are being listed below.…”

“…Upon setting y = 0 and b = c = 0, the general q-polynomials L(r,s) n (α, x, y, z, a, b, c) defined in (12) would reduce to (8) (see [10])…”