Asymptotics of Saran's hypergeometric function $F_K$

TL;DR

This paper derives asymptotic expansions for Saran's hypergeometric function $F_K$ by analyzing the Humbert function $ ext{ extPsi}_1$ for large variables, providing insights into its behavior in asymptotic regimes.

Contribution

It introduces new asymptotic expansions for $F_K$ when two variables are large, based on expansions of the Humbert function $ ext{ extPsi}_1$, advancing understanding of hypergeometric functions.

Findings

Asymptotic expansion of $ ext{ extPsi}_1$ for large variables

Asymptotic expansion of $F_K$ for two large variables

Enhanced understanding of hypergeometric function behavior in asymptotic limits

Abstract

In this paper, we first establish asymptotic expansions of the Humbert function $\Psi_1$ for one large variable. The resulting expansions are then used to derive an asymptotic expansion of Saran's hypergeometric function $F_K$ when two of its variables become simultaneously large.