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

Authors : Qi Bao, Dunkun Yang

Pages : 59-76

Doi:10.33401/fujma.1365120

View : 63 | Download : 68

Publication Date : 2024-06-30

Article Type : Research

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