Liangjian Hu | Searcharxiv

arXiv2026

Asymptotic expansions of the Humbert Function $Φ_1$ and their applications

This paper systematically studies the asymptotics of Humbert's bivariate confluent hypergeometric function $Φ_1[a,b;c;x, y]$. Specifically, we establish explicit asymptotic expansions in five distinct regimes: (i) $x\to\infty$; (ii) $y\to\infty$; (iii) $x\to\infty,\,y\to\infty$; (iv) $x$ or $y$ small, $xy$ fixed; and (v) $x\to 1$, $y$ fixed. The utility of these expansions is illustrated through concrete applications in the theory of Saran's hypergeometric function $F_M$, the Glauber-Ising model, and the theory of Prabhakar-type fractional integral operators. Several potential directions for future work are also outlined.

math.CA↗

arXiv2025

Complete asymptotic expansions of the Humbert function $Ψ_1$ for two large arguments

In our recent work [SIGMA \textbf{20} (2024), 074, 13 pages], the leading behaviour of the Humbert function $Ψ_1[a,b;c,c';x,y]$ 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 $Ψ_1$ 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 $z$.

math.CA↗