On the Heine Binomial Operators

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 $\Phi_{m}^{(q^n)}(b,x|q)$.