On integral representations and asymptotics of some hypergeometric functions in two variables

TL;DR

This paper derives asymptotic behaviors and new integral representations for Humbert functions of two variables, using inverse Laplace transforms and Tauberian theorems to analyze their growth at large arguments.

Contribution

It introduces novel integral representations of Humbert functions and applies Tauberian theorems to determine their asymptotic behavior in two variables.

Findings

Asymptotic formulas for Humbert functions when variables are large

New integral representations as inverse Laplace transforms

Analysis of specific integrals involving Humbert functions

Abstract

The leading asymptotic behaviour of the Humbert functions $\Phi_2$, $\Phi_3$, $\Xi_2$ of two variables is found, when the absolute values of the two independent variables become simultaneosly large. New integral representations of these functions are given. These are re-expressed as inverse Laplace transformations and the asymptotics is then found from a Tauberian theorem. Some integrals of the Humbert functions are also analysed.