Asymptotics of a ${}_3F_2$ hypergeometric function with four large parameters

TL;DR

This paper investigates the asymptotic behavior of a specific generalized hypergeometric function with four large parameters as the parameter k approaches infinity, providing numerical validation of the derived expansion.

Contribution

It derives an asymptotic expansion for a ${}_3F_2$ hypergeometric function with four large parameters and confirms its accuracy through numerical experiments.

Findings

Asymptotic expansion for the hypergeometric function is obtained.

Numerical results validate the accuracy of the asymptotic approximation.

The expansion is effective for large parameter k.

Abstract

We consider the asymptotic behaviour of the generalised hypergeometric function \[{}_3F_2\bl(\!\!\begin{array}{c} 1, (1+t)k/2, (1+t)k/2+1/2\\tk+1, k+1\end{array}\!\!; x\br),\qquad 0<x,t\leq 1\] as the parameter $k\to+\infty$. Numerical results illustrating the accuracy of the resulting expansion are given.