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

Zhong-Xuan Mao; Jing-Feng Tian

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arXiv · 2603.16465

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

Zhong-Xuan Mao·Jing-Feng Tian

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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 $\arccos(pz)$.

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Zhong-Xuan Mao, Jing-Feng Tian. 2026-03-17. Recurrence relations for the Maclaurin coefficients of products of elementary functions and Hypergeometric functions. https://arxiv.org/abs/2603.16465

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