Formulas for Poisson–Charlier, Hermite, Milne-Thomson and other type polynomials by their generating functions and p-adic integral approach

Abstract

The main propose of this article is to investigate and modify Hermite type polynomials, Milne-Thomson type polynomials and Poisson–Charlier type polynomials by using generating functions and their functional equations. By using functional equations of the generating functions for these polynomials, we not only derive some identities and relations including the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Poisson–Charlier polynomials, the Milne-Thomson polynomials and the Hermite polynomials, but also study some fundamental properties of these functions and polynomials. Moreover, we survey orthogonality properties of these polynomials. Finally, by applying another method which is related to p-adic integrals, we derive some formulas and combinatorial sums associated with some well-known numbers and polynomials.