Closed form expressions for Appell polynomials

TL;DR

This paper derives explicit closed-form expressions for Appell polynomials using forward difference transformations, revealing their structure and convolution identities, with applications to generalized Bernoulli and Apostol-Euler polynomials.

Contribution

It introduces a novel closed-form representation of Appell sequences as forward difference transformations, linking them to Stirling numbers and convolution identities.

Findings

Explicit formulas for Appell polynomials derived

Convolution identities expressed via Stirling numbers

Applications to generalized Bernoulli and Apostol-Euler polynomials

Abstract

We show that any Appell sequence can be written in closed form as a forward difference transformation of the identity. Such transformations are actually multipliers in the abelian group of the Appell polynomials endowed with the operation of binomial convolution. As a consequence, we obtain explicit expressions for higher order convolution identities referring to various kinds of Appell polynomials in terms of the Stirling numbers. Applications of the preceding results to generalized Bernoulli and Apostol-Euler polynomials of real order are discussed in detail.