The generalized q-heat equations for q-3D hypergeometric polynomials with applications to generating functions and Askey–Wilson integrals

Abstract

Polynomial expansions of analytic solutions of the heat equation occupy important positions in disciplines such as mathematics and physics [68]. In this paper, we introduce q-3D hypergeometric polynomials and find their corresponding q-heat equations, which were motivated by Ismail and Zhang (2016) [32] and (2017) [33]. We deduce several types of generating functions for q-3D hypergeometric polynomials and Askey–Wilson type integral involving q-3D hypergeometric polynomials by the method of heat equation type q-partial differential equations. In addition, we generalize some results of Ismail and Zhang (2017) [33], Milne (1997) [49] and Jia (2021) [38].