Two q-Operational Equations and Hahn Polynomials

Abstract

Motivated by Liu’s (Sci China Math 66:1199–1216, 2023) recent work. This article reveals the essential features of Hahn polynomials by presenting a new q-exponential operator, that is $$\begin{aligned} \exp _q(t\Delta _{x,a})f(x)=\frac{(axt;q)_{\infty }}{(xt;q)_{\infty }} \sum _{n=0}^{\infty }\frac{t^n}{(q;q)_n} f(q^n x) \end{aligned}$$ with \(\Delta _{x,a}=x (1-a)\eta _a+\eta _x\) and \(\eta _x \{f(x) \}=f(qx)\) . Letting \(f(x) \equiv 1\) and the above operator equation immediately becomes the generating function of Hahn polynomials. These lead us to use a systematic method for studying identities involving Hahn polynomials. As applications, we use the method of the q-exponential operator to prove some new q-identities, including q-Nielsen’s formulas and Carlitz’s extension for the Hahn polynomials, etc. Moreover, a generalization of q-Gauss summation is given, too.