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

Abstract

We consider the class 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 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.

A probabilistic generalization of the Stirling numbers of the second kind

“…where in the last equality we have used (23). This, together, with (30), shows (31) and completes the proof.…”

“…Despite the cumbersome expression in (20), if we replace each y j by the random variable Y j , as given in (7), and then take expectations, we obtain the following auxiliary result already shown in [23…”

“…The following result, which generalizes in various ways Proposition 2.2 in [23], gives us equivalent definitions of the polynomials S Y (n, m; x). In this respect, recall that the classical Stirling numbers of the first and second kind, respectively denoted by s(n, k) and S(n, k), k = 0, 1, .…”

“…Choosing f (x) = I n (x) in (26) and recalling (23), we see that the definition of the Stirling polynomials of the second kind associated to Y , given in (8), makes sense. In other words,…”

Closed form expressions for Appell polynomials

“…Section 2 contains some basic results on binomial convolutions of Appell sequences already shown in [3]. In Section 3, we introduce the notion of forward difference transformation of an Appell sequence, so that formula (2) states that any Appell sequence A(x) is a forward difference transformation of the identity. We have also included in Section 4 some properties of the Stirling numbers, because the sequence a = (a n ) n≥0 may be described in terms of them.…”