Full asymptotic expansions of the Humbert function $Φ_1$

Abstract: We derive full asymptotic expansions for the Humbert function $\Phi_1$ in different limiting regimes of its variables. Our derivation employs various asymptotic methods and relies on key transformation formulae established by Erd\'elyi (1940), and Tuan and Kalla (1987). The efficiency of our asymptotic results are also illustrated through two applications: (1) analytic continuations of Saran's function $F_M$, and (2) two limits arising in the study of the $1D$ Glauber-Ising model. Finally, some promising directions for future research are highlighted.