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

In our recent work [SIGMA $\mathbf{\text{20}}$ (2024), 074, 13 pages], the leading behaviour of the Humbert function ${\mathrm{\Psi }}_1[a,b;c,c^{\mathrm{\prime }};x,y]$ when $x\to \mathrm{\infty }$ and $y\to +\mathrm{\infty }$ has been derived in a direct and simple manner. In this paper, we obtain the complete asymptotics of ${\mathrm{\Psi }}_1$ in the general case $x,y\to \mathrm{\infty }$ 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 $z$.