Recurrence relations for the Maclaurin coefficients of products of elementary functions and Hypergeometric functions

TL;DR

This paper derives recurrence relations for Maclaurin coefficients of products involving elementary functions and hypergeometric functions, providing a systematic way to compute these coefficients for various function combinations.

Contribution

It introduces new recurrence relations for Maclaurin coefficients of products of elementary functions with hypergeometric functions, covering multiple specific cases of $h(z)$.

Findings

Derived recurrence relations for specific $h(z)$ functions

Facilitated computation of Maclaurin coefficients for complex products

Extended understanding of hypergeometric function expansions

Abstract

In this paper, we investigate the recurrence relations for the Maclaurin coefficients of the products of elementary functions and hypergeometric functions. Specifically, we focus on the confluent hypergeometric function $\mathcal{M}(z) = h(z) M(a,c;z)$ and the Gaussian hypergeometric function $\mathcal{F}(z) = h(z) F(a,b;c;z)$, considering several specific choices for the function $h(z)$. In particular, we explore cases where $h(z)$ is chosen as $e^{pz}$, $(1-\theta z)^p$, $e^{-p \arctan z}$, $\sin(pz)$, $\cos(pz)$, $\sinh(pz)$, $\cosh(pz)$, $\arcsin(pz)$, and $\arccos(pz)$.