A general method to find special functions that interpolate Appell polynomials, with examples

Abstract

Given an Appell sequence {Pn(x)}n=0∞ defined by means of a generating functionA(t)ext=∑n=0∞Pn(x)tnn!, we discuss a general procedure for constructing a complex function F(s,x), which is entire in s for each fixed x with Rex>0, and satisfies F(−n,x)=Pn(x) at n=0,1,2,…. The method is based on the Mellin transform and allows A(−t) to have isolated singularities on the half-line (0,∞), in contrast with other general methods that appear in the mathematical literature. We illustrate our procedure with some elucidatory examples. However, our approach cannot be used for analogously defined Appell-Dunkl sequences, a fact which has led us to include an open problem related to this case.