q-Difference equation and the Cauchy operator identities

Abstract

Basic hypergeometric series q-Differential operator The Cauchy operator Multiple basic hypergeometric series In this paper, we verify the Cauchy operator identities by a new method. And by using the Cauchy operator identities, we obtain a generating function for Rogers-Szegö polynomials. Applying the technique of parameter augmentation to two multiple generalizations of q-Chu-Vandermonde summation theorem given by Milne, we also obtain two multiple generalizations of the Kalnins-Miller transformation.

On a Reduction Formula for a Kind of Double q-Integrals

“…On making use of the q-exponential operator method, we [3] proved the following proposition (some misprints have been corrected here): Proposition 1. For max{|au|, |bu|, |cu|, |av|, |bv|, |cv|, |d/b|} < 1, we have: When d = abcuv in Proposition 1, upon noting that (1; q) k = δ 0,k , the 3 φ 2 series in the proposition reduces to 1 and thus the proposition becomes the Al-Salam-Verma formula, which is the q-integral representation of Sears' nonterminating extension of the q-Saalschütz summation [4], see also [5] (p. 52).…”

q-Difference equation and q-polynomials

“…[18]), q-operator method (cf. [5][6][7][8][9][10][11][12][13][20][21][22]25]) and so on. Inspired by [7,8,10,[20][21][22], in this paper, we shall use q-difference equation to discuss some proprieties of H n defined as follows: …”

“…For instance, the authors [10,20,21,25] made a systematic study on Fð0; À; cD q;b Þ. Some applications of Fða 0 ; À; cD q;b Þ were given in [11,12,22].…”

“…For instance, the authors [10,20,21,25] made a systematic study on Fð0; À; cD q;b Þ. Some applications of Fða 0 ; À; cD q;b Þ were given in [11,12,22]. And some properties and applications of Fða 0 ; a 1 ; b 1 ; cD q;b Þ were discussed in [7,12].…”

Generalizations of certain Carlitz’s trilinear and Srivastava–Agarwal type generating functions

“…The author of[10,15,16] derived several generating functions for Rogers-Szegö polynomials by q-exponential operators. For more information about Rogers-Szegö and Hahn polynomials, please refer to[7,8,[10][11][12][14][15][16][17][18][19][20][21][22]. Carlitz [7, Eq.…”