Convolution identities for scale transformations of Appell polynomials

TL;DR

This paper derives closed-form convolution identities for a subset of Appell polynomials, including Bernoulli and Apostol-Euler, using a probabilistic approach based on Stirling numbers, unifying and extending existing results.

Contribution

It introduces a unified probabilistic method to derive convolution identities for a class of Appell polynomials, encompassing several well-known polynomial families.

Findings

Closed-form convolution identities for Appell polynomials derived

Unified approach based on probabilistic Stirling numbers

Reformulations of known identities and new generalizations

Abstract

We obtain closed form expressions for convolutions of scale transformations within a certain subset of Appell polynomials. This subset contains the Bernoulli, Apostol-Euler, and Cauchy polynomials, as well as various kinds of their generalizations, among others. We give a unified approach mainly based 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.