Uniform asymptotics of a Gauss hypergeometric function with two large parameters, V

TL;DR

This paper derives uniform asymptotic expansions for the Gauss hypergeometric function with two large parameters near a critical point, using two different methods to improve understanding of its behavior.

Contribution

It introduces two novel uniform asymptotic expansions for the hypergeometric function with large parameters, employing Bleistein's method and Olver's approach, with advantages over previous methods.

Findings

Derived a uniform asymptotic expansion using Bleistein's method.

Presented an alternative form of the expansion based on Olver's approach.

Compared the two forms, highlighting their respective advantages.

Abstract

We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[{}_2F_1(a+\epsilon\lambda,b;c+\lambda;x),\qquad 0<x<1\] as $\lambda\to+\infty$ in the neigbourhood of $\epsilon x=1$ when the parameter $\epsilon>1$ and the constants $a$, $b$ and $c$ are supposed finite. Use of a standard integral representation shows that the problem reduces to consideration of a simple saddle point near an endpoint of the integration path. A uniform asymptotic expansion is first obtained by employing Bleistein's method. An alternative form of uniform expansion is derived following the approach described in Olver's book [{\it Asymptotics and Special Functions}, p.~346]. This second form has several advantages over the Bleistein form.