A conjecture of Gasper on $q$-ultraspherical polynomials

TL;DR

This paper proves a $q$-orthogonality relation for continuous $q$-ultraspherical polynomials and evaluates a new multi-parameter $q$-beta integral, advancing the understanding of $q$-special functions.

Contribution

It establishes the $q$-orthogonality relation for $q$-ultraspherical polynomials and derives a novel $q$-beta integral with multiple parameters, addressing Gasper's conjecture.

Findings

Proved $q$-orthogonality for continuous $q$-ultraspherical polynomials.

Derived a new multi-parameter $q$-beta integral.

Enhanced the theoretical framework of $q$-special functions.

Abstract

In this paper we establish $q$-orthogonality relation for the continuous $q$-ultraspherical polynomials, which was considered by Gasper. Additionally, we evaluate a new $q$-beta integral with several parameters.