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

Based on the extensive application of the $q$-series and $q$-polynomials including (for example) the $q$-Laguerre polynomials in several fields of the mathematical and physical sciences, we attach great importance to the equations and related application issues involving the $q$-Laguerre polynomials. The mission of this paper is to find the general $q$-operational equation together with the expansion issue of the bivariate $q$-Laguerre polynomials from the perspective of $q$-partial differential equations. We also give some applications including some $q$-Hille-Hardy type formulas. In addition, we present the Rogers-type formulas and the $U(n+1)$-type generating functions for the bivariate $q$-Laguerre polynomials by the technique based upon $q$-operational equations. Moreover, we derive a new generalized Andrews-Askey integral and a new transformation identity involving the bivariate $q$-Laguerre polynomials by applying $q$-operational equations. $U(n+1)$