A $q$-summation and the orthogonality relations for the $q$-Hahn polynomials and the big $q$-Jacobi polynomials

TL;DR

This paper introduces a new $q$-summation formula that facilitates deriving orthogonality relations for $q$-Hahn and big $q$-Jacobi polynomials, along with new integral formulas and identities.

Contribution

The paper presents a novel $q$-summation formula that simplifies proofs of orthogonality and integrals for specific $q$-polynomials, advancing the theoretical understanding of $q$-series.

Findings

Derived a generating function for $q$-Hahn polynomials

Provided a new proof of orthogonality for big $q$-Jacobi polynomials

Established a new $q$-beta integral formula

Abstract

Using a general $q$-summation formula, we derive a generating function for the $q$-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the $q$-Hahn polynomials. A new proof of the orthogonality relation for the big $q$-Jacobi polynomials is also given. A simple evaluation of the Nassrallah-Rahman integral is derived by using this summation formula. A new $q$-beta integral formula is established, which includes the Nassrallah-Rahman integral as a special case. The $q$-summation formula also allows us to recover several strange $q$-series identities.