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

2022/07/04 by Qi Bao, Bao, Qi, Yang, DunKun
Chemistry · Mathematics · Medicine · #05A30 #11B65 #32A05 #33D15 #33D45 #39A13 #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #Drug Transport and Resistance Mechanisms #FOS: Mathematics #Functional Analysis (math.FA) #Molecular spectroscopy and chirality

paper · pdf · doi:10.48550/arxiv.2207.01442

openalex publication_date 2022/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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