Quantum (or q$$ q $$‐) operator equations and associated partial differential equations for bivariate Laguerre polynomials with applications to the q$$ q $$‐Hille‐Hardy type formulas

Abstract

Based on the extensive application of the ‐series and ‐polynomials including (for example) the ‐Laguerre polynomials in several fields of the mathematical and physical sciences, we attach great importance to the equations and related application issues involving the ‐Laguerre polynomials. The mission of this paper is to find the general ‐operational equation together with the expansion issue of the bivariate ‐Laguerre polynomials from the perspective of ‐partial differential equations. We also give some applications including some ‐Hille‐Hardy type formulas. In addition, we present the Rogers‐type formulas and the ‐type generating functions for the bivariate ‐Laguerre polynomials by the technique based upon ‐operational equations. Moreover, we derive a new generalized Andrews‐Askey integral and a new transformation identity involving the bivariate ‐Laguerre polynomials by applying ‐operational equations.