Two $q$-operational equations and Hahn polynomials

TL;DR

This paper introduces two new $q$-exponential operators to systematically study Hahn polynomials, proving their generating functions, transformation formulas, and generalizing $q$-Gaussian summation.

Contribution

The paper presents novel $q$-exponential operators that reveal key features of Hahn polynomials and facilitate derivation of important identities and generalizations.

Findings

Proved the bilinear generating function of Hahn polynomials.

Derived Heine's second transformation formula using $q$-exponential operators.

Generalized $q$-Gaussian summation.

Abstract

Motivated by Liu's recent work in \cite{Liu2022}. We shall reveal the essential feature of Hahn polynomials by presenting two new $q$-exponential operators. These lead us to use a systematic method to study identities involving Hahn polynomials. As applications, we use the method of $q$-exponential operator to prove the bilinear generating function of Hahn polynomials and Heine's second transformation formula. Moreover, a generalization of $q$-Gaussian summation is given, too.