  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.
