Two q-Operational Equations and Hahn Polynomials

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 \(\begin{aligned} \exp _q(t\Delta _{x,a})f(x)=\frac{(axt;q)_{\infty }}{(xt;q)_{\infty }} \sum _{n=0}^{\infty }\frac{t^n}{(q;q)_n} f(q^n x) \end{aligned}\) $$\begin{array}{c}\hfill {exp}_q(t{\mathrm{\Delta }}_{x,a})f(x)=\frac{{(axt;q)}_{\infty }}{{(xt;q)}_{\infty }}\sum _{n=0}^{\infty }\frac{t^n}{{(q;q)}_n}f(q^{n}x)\end{array}$$ with \(\Delta _{x,a}=x (1-a)\eta _a+\eta _x\) ${\mathrm{\Delta }}_{x,a}=x(1-a){\eta }_a+{\eta }_x$ and \(\eta _x \{f(x) \}=f(qx)\) ${\eta }_x\{f(x)\}=f(qx)$ . Letting \(f(x) \equiv 1\) $f(x)\equiv 1$ 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.