Notes on $q$-Partial Differential Equations for $q$-Laguerre Polynomials and Little $q$-Jacobi Polynomials

Abstract

This article defines 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. Furthermore, 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, several generalized Ramanujan $q$-beta integrals and Andrews-Askey integrals are obtained.

Keywords

• $q$-Laguerre polynomial
• little $q$-Jacobi polynomial
• $q$-partial differential equations
• generating function
• Ramanujan $q$-beta integrals
• $q$-integrals

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