Asymptotics of a Gauss hypergeometric function with large parameters, III: Application to the Legendre functions of large imaginary order and real degree

TL;DR

This paper derives asymptotic expansions for a specific hypergeometric function with large parameters, applying steepest descent, and explores their application to Legendre functions of large imaginary order, including numerical validation.

Contribution

It provides new asymptotic formulas for hypergeometric functions with large parameters, specifically applied to Legendre functions of large imaginary order, using steepest descent methods.

Findings

Derived Poincaré-type asymptotic expansions for the hypergeometric function.

Provided expansions at saddle point coalescence.

Numerical results confirm the accuracy of the expansions.

Abstract

We obtain the asymptotic expansion for the Gauss hypergeometric function \[F(a-\lambda,b+\lambda;c+i\alpha\lambda;z)\] for $\lambda\rightarrow+\infty$ with $a$, $b$ and $c$ finite parameters by application of the method of steepest descents. The quantity $\alpha$ is real, so that the denominatorial parameter is complex and $z$ is a finite complex variable restricted to lie in the sector $|\arg (1-z)|<\pi$. We concentrate on the particular case $a=0$, $b=c=1$, which is associated with the Legendre functions of real degree and imaginary order. The resulting expansions are of Poincar\'e type and hold in restricted domains of the $z$-plane. An expansion is given at the coalescence of two saddle points. Numerical results illustrating the accuracy of the different expansions are given.