A Unified Approach to Higher Order Convolutions Within a Certain Subset of Appell Polynomials

Abstract

We consider the subset R of Appell polynomials whose exponential generating function is given in terms of the moment generating function of a certain random variable Y . This subset contains the Hermite, Bernoulli, Apostol-Euler, and Cauchy type polynomials, as well as various kinds of their generalizations, among others. We obtain closed form expressions for higher-order convolutions of Appell polynomials in the subset R. We give a unified approach mainly based on random scale transformations of Appell polynomials, as well as on a probabilistic generalization of the Stirling numbers of the second kind. Different illustrative examples, including reformulations of convolution identities already known in the literature, are discussed in detail. In such examples, the convolution identities involve the classical Stirling numbers.

Probabilistic Stirling Numbers of the Second Kind and Applications

“…where β(r +1) is the random variable defined in (4). Replacing z by zY in this formula and then taking expectations, we have from (5) and 111 m!…”

“…The motivations behind definition (3) have to do with certain problems coming from analytic number theory, such as extensions in various ways of the classical formula for sums of powers on arithmetic progressions (cf. [3]) and explicit expressions for higher-order convolutions of Appell polynomials (see [4]).…”

Probabilistic Stirling numbers of the second kind and applications

“…where β(r + 1) is the random variable defined in (4). Replacing z by zY in this formula and then taking expectations, we have from ( 5) and ( 11)…”

“…The motivations behind definition (3) have to do with certain problems coming from analytic number theory, such as extensions in various ways of the classical formula for sums of powers on arithmetic progressions (cf. [3]) and explicit expressions for higher order convolutions of Appell polynomials (see [4]).…”