Homogeneous q-partial difference equations and some applications

In this paper, we show how to prove identities and evaluate integrals by expanding functions in terms of products of the q-hypergeometric polynomials by homogeneous q-partial difference equations, we also generalize some results of Liu (2015) [34] and Cao (2016) [14]. In addition, we generalize multilinear and multiple generating functions for the q-hypergeometric polynomials as applications. Moreover, we deduce some recurring formulas for Ramanujan's integrals, Askey-Roy integrals, Andrews-Askey integrals and moment integrals by the method of homogeneous q-partial difference equations. Finally, we build the relation of Ismail-Zhang type generating functions for the q-hypergeometric polynomials by the method of Ismail and Zhang (2016) [26].