Some Expansion Formulas for Brenke Polynomial Sets

TL;DR

This paper derives explicit expansion formulas for Brenke polynomial sets using generating functions, providing new coefficients expressions and applying results to generalized polynomials, including classical duplication and convolution formulas.

Contribution

It introduces novel explicit expansion formulas for Brenke polynomials and their applications to generalized polynomial families, enhancing computational and theoretical understanding.

Findings

Explicit formulas expressed as finite sums or hypergeometric terms

Applications to generalized Gould-Hopper and Hermite polynomials

Derivation of classical duplication and convolution formulas

Abstract

In this paper, we derive some explicit expansion formulas associated to Brenke polynomials using operational rules based on their corresponding generating functions. The obtained coefficients are expressed either in terms of finite double sums or finite sums or sometimes in closed hypergeometric terms. The derived results are applied to Generalized Gould-Hopper polynomials and Generalized Hermite polynomials introduced by Szeg\"{o} and Chihara. Some well-known duplication and convolution formulas are deduced as particular cases.