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 exp(q)(t Delta(x,a)) f(x) = (axt;q)infinity/(xt;q)infinity & sum;(infinity)(n=0 )t(n)/(q;q)(n )f (q(n)x) with Delta(x,a)=x(1-a)eta(a)+eta(x )and eta(x){f(x)}=f(qx). Letting f(x)equivalent to 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.