Notes on $q$-partial differential equations for $q$-Laguerre polynomials and little $q$-Jacobi polynomials
Classical Analysis and ODEs
2023-05-09 v3 Functional Analysis
Abstract
We define two common -orthogonal polynomials: homogeneous -Laguerre polynomials and homogeneous little -Jacobi polynomials. They can be viewed separately as solutions to two -partial differential equations. Then, we proved that if an analytic function satisfies a certain system of -partial differential equations, if and only if it can be expanded in terms of homogeneous -Laguerre polynomials or homogeneous little -Jacobi polynomials. As applications, we obtain generalizations of the Ramanujan -beta integrals and Andrews-Askey integrals. Additionally, we present an operator representation of -Laguerre polynomials that facilitates the computation of identities involving -Laguerre polynomials.
Keywords
Cite
@article{arxiv.2207.01442,
title = {Notes on $q$-partial differential equations for $q$-Laguerre polynomials and little $q$-Jacobi polynomials},
author = {Qi Bao and DunKun Yang},
journal= {arXiv preprint arXiv:2207.01442},
year = {2023}
}