Appell polynomials as values of special functions

We show that there is a large class of Appell sequences {Pn(x)}n=0∞ for which there is a function F(s,x), entire in s for fixed x with Rex>0 and satisfying F(−n,x)=Pn(x) for n=0,1,2,…. For example, in the case of Bernoulli and Apostol–Bernoulli polynomials, F is essentially the Hurwitz zeta function and the Lerch transcendent, respectively. We study a subclass of these Appell sequences for which the corresponding special function has a form more closely related to the classical zeta functions, and give some interesting examples of these general constructions.