On the Heine Binomial Operators

In this paper, we introduce the Heine binomial operators H${}_n(b{D}_q)$ based on $q$-differential operator $D_q$. The motivation for introducing the operators H${}_n(b{D}_q)$ is that their limit turns out to be the $q$-exponential operator T$(b{D}_q)$ given by Chen. The Hahn polynomials ${\mathrm{\Phi }}_m^{(q^n)}(b,x\mathrm{\mid }q)$ can easily be represented by using the operators H${}_n(b{D}_q)$. Here, we derive $q$-exponential and ordinary generating function, Mehler’s formula, Rogers formula, and other identities for the polynomials ${\mathrm{\Phi }}_m^{(q^n)}(b,x\mathrm{\mid }q)$.