Asymptotic expansions of the Humbert Function $\Phi_1$ and their applications

This paper systematically studies the asymptotics of Humbert's bivariate confluent hypergeometric function $\Phi_1[a,b;c;x, y]$. 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.