Notes on $q$-partial differential equations for $q$-Laguerre polynomials and little $q$-Jacobi polynomials

We define two common $q$-orthogonal polynomials: homogeneous $q$-Laguerre polynomials and homogeneous little $q$-Jacobi polynomials. They can be viewed separately as solutions to two $q$-partial differential equations. Then, we proved that if an analytic function satisfies a certain system of $q$-partial differential equations, if and only if it can be expanded in terms of homogeneous $q$-Laguerre polynomials or homogeneous little $q$-Jacobi polynomials. As applications, we obtain generalizations of the Ramanujan $q$-beta integrals and Andrews-Askey integrals. Additionally, we present an operator representation of $q$-Laguerre polynomials that facilitates the computation of identities involving $q$-Laguerre polynomials.