A $q$-linear analogue of the plane wave expansion

TL;DR

This paper develops a $q$-linear analogue of Gegenbauer's plane wave expansion, expressing it through little $q$-Gegenbauer polynomials and the third Jackson $q$-Bessel function, using bilinear biorthogonal expansions.

Contribution

It introduces a novel $q$-linear analogue of the classical plane wave expansion, expanding it in terms of $q$-special functions with a new method.

Findings

Derived a $q$-analogue of Gegenbauer's expansion

Expressed the expansion using little $q$-Gegenbauer polynomials

Utilized bilinear biorthogonal expansions for the derivation

Abstract

We obtain a $q$-linear analogue of Gegenbauer's expansion of the plane wave. It is expanded in terms of the little $q$-Gegenbauer polynomials and the \textit{third} Jackson $q$-Bessel function. The result is obtained by using a method based on bilinear biorthogonal expansions.