Complete asymptotic expansions of the Humbert function ${\Psi }_1$ for two large arguments

Abstract

In our recent work [SIGMA \textbf{20} (2024), 074, 13 pages], the leading behaviour of the Humbert function ${\Psi }_1[a,b;c,c^{\prime };x,y]$ 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 ${}_2{F}_2[a,b-n;c,d-n;z]$, which is valid uniformly for $n\in {\mathbb{Z}}_{⩾0}$ and large $z$.

10 pages, 0 figures