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

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