On the Heine Binomial Operators

TL;DR

This paper introduces the Heine binomial operators based on q-differential operators, explores their limits, and derives various generating functions and identities for associated Hahn polynomials.

Contribution

It defines new Heine binomial operators linked to q-differential operators and derives key identities and generating functions for related Hahn polynomials.

Findings

Heine binomial operators generalize q-exponential operators

Derived generating functions for Hahn polynomials

Established identities like Mehler's and Rogers formulas

Abstract

In this paper, we introduce the Heine binomial operators H$_{n}(bD_{q})$ based on $q$-differential operator $D_{q}$. The motivation for introducing the operators H$_{n}(bD_{q})$ is that their limit turns out to be the $q$-exponential operator T$(bD_{q})$ given by Chen. The Hahn polynomials $\Phi_{m}^{(q^n)}(b,x|q)$ can easily be represented by using the operators H$_{n}(bD_{q})$. Here, we derive $q$-exponential and ordinary generating function, Mehler's formula, Rogers formula, and other identities for the polynomials $\Phi_{m}^{(q^n)}(b,x|q)$.