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

TL;DR

This paper introduces two $q$-orthogonal polynomial families as solutions to specific $q$-partial differential equations, providing new integral generalizations and operator representations that enhance understanding and computation of related identities.

Contribution

It defines homogeneous $q$-Laguerre and little $q$-Jacobi polynomials via $q$-partial differential equations and offers new integral formulas and operator methods for these polynomials.

Findings

Characterization of $q$-Laguerre and little $q$-Jacobi polynomials as PDE solutions

Generalizations of Ramanujan $q$-beta and Andrews-Askey integrals

Operator representation aiding polynomial identity computations

Abstract

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.