Complete asymptotic expansions of the Humbert function $\Psi_1$ for two large arguments

TL;DR

This paper derives complete asymptotic expansions for the Humbert function  in the case of two large arguments, extending previous results by analyzing the function along a new path using sharp estimates.

Contribution

It provides the first complete asymptotic expansions of  for two large arguments along a new path, based on uniform estimates of hypergeometric functions.

Findings

Derived complete asymptotics of  for large arguments.

Introduced a new method using sharp estimates of hypergeometric functions.

Extended previous asymptotic results to a more general case.

Abstract

In our recent work [SIGMA \textbf{20} (2024), 074, 13 pages], the leading behaviour of the Humbert function $\Psi_1[a,b;c,c';x,y]$ when $x\to\infty$ and $y\to +\infty$ has been derived in a direct and simple manner. In this paper, we obtain the complete asymptotics of $\Psi_1$ in the general case $x,y\to\infty$ along a new path. Indeed, our proof is based on a sharp estimate on ${}_2F_2[a,b-n;c,d-n;z]$, which is valid uniformly for $n\in\mathbb{Z}_{\geqslant 0}$ and large $z$.