English

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

Complex Variables 2026-03-18 v1

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 M(z)=h(z)M(a,c;z)\mathcal{M}(z) = h(z) M(a,c;z) and the Gaussian hypergeometric function F(z)=h(z)F(a,b;c;z)\mathcal{F}(z) = h(z) F(a,b;c;z), considering several specific choices for the function h(z)h(z). In particular, we explore cases where h(z)h(z) is chosen as epze^{pz}, (1θz)p(1-\theta z)^p, eparctanze^{-p \arctan z}, sin(pz)\sin(pz), cos(pz)\cos(pz), sinh(pz)\sinh(pz), cosh(pz)\cosh(pz), arcsin(pz)\arcsin(pz), and arccos(pz)\arccos(pz).

Keywords

Cite

@article{arxiv.2603.16465,
  title  = {Recurrence relations for the Maclaurin coefficients of products of elementary functions and Hypergeometric functions},
  author = {Zhong-Xuan Mao and Jing-Feng Tian},
  journal= {arXiv preprint arXiv:2603.16465},
  year   = {2026}
}