Explicit expressions for a certain class of Appell polynomials. A probabilistic approach

TL;DR

This paper introduces explicit formulas for a class of Appell polynomials using a probabilistic approach, unifying various generalizations of Bernoulli and Apostol-Euler polynomials through moments of a random variable.

Contribution

It provides explicit probabilistic expressions for Appell polynomials in the class \mathcal{E}_t(Y), linking them to moments and Stirling numbers, and unifies their analysis.

Findings

Explicit formulas depending on moments of Y

Unified treatment of Bernoulli and Apostol-Euler generalizations

Probabilistic Stirling numbers for polynomial expressions

Abstract

We consider the class $\mathcal{E}_t(Y)$ of Appell polynomials whose generating function is given by means of a real power $t$ of the moment generating function of a certain random variable $Y$. For such polynomials, we obtain explicit expressions depending on the moments of $Y$. It turns out that various kinds of generalizations of Bernoulli and Apostol-Euler polynomials belong to $\mathcal{E}_t(Y)$ and can be written and investigated in a unified way. In particular, explicit expression for such polynomials can be given in terms of suitable probabilistic generalizations of the Stirling numbers of the second kind.