On (self-) reciprocal Appell polynomials: Symmetry and Faulhaber-type polynomials

TL;DR

This paper explores the symmetry properties of generalized Appell polynomials, revealing their connection to Faulhaber-type polynomials and deriving applications to Bernoulli and Euler polynomials.

Contribution

It establishes a link between reciprocal Appell polynomials and Faulhaber-type polynomials via quadratic substitution, with explicit coefficient formulas and applications.

Findings

Appell polynomials satisfying reflection relations can be described by Faulhaber-type polynomials.

Coefficients of Faulhaber-type polynomials are derivatives of reciprocal Appell polynomials.

Results apply to classical power sums and Faulhaber polynomials.

Abstract

The main purpose of this paper is to study generalized (self-) reciprocal Appell polynomials, which play a certain role in connection with Faulhaber-type polynomials. More precisely, we show for any Appell sequence when satisfying a reflection relation that the Appell polynomials can be described by Faulhaber-type polynomials, which arise from a quadratic variable substitution. Furthermore, the coefficients of the latter polynomials are given by values of derivatives of generalized reciprocal Appell polynomials. Subsequently, we show some applications to the Bernoulli and Euler polynomials. In the context of power sums the results transfer to the classical Faulhaber polynomials.