The asymptotics of a generalised Beta function

TL;DR

This paper derives the asymptotic behavior of a generalized Beta function under various large-parameter limits, providing formulas and numerical validation for these asymptotics.

Contribution

It presents new asymptotic formulas for the generalized Beta function in multiple large-parameter regimes, extending previous understanding.

Findings

Asymptotic formulas for large p with fixed x and y

Asymptotic formulas for large x and p

Numerical validation of derived asymptotics

Abstract

We consider the generalised Beta function introduced by Chaudhry {\it et al.\/} [J. Comp. Appl. Math. {\bf 78} (1997) 19--32] defined by \[B(x,y;p)=\int_0^1 t^{x-1} (1-t)^{y-1} \exp \left[\frac{-p}{4t(1-t)}\right]\,dt,\] where $\Re (p)>0$ and the parameters $x$ and $y$ are arbitrary complex numbers. The asymptotic behaviour of $B(x,y;p)$ is obtained when (i) $p$ large, with $x$ and $y$ fixed, (ii) $x$ and $p$ large, (iii) $x$, $y$ and $p$ large and (iv) either $x$ or $y$ large, with $p$ finite. Numerical results are given to illustrate the accuracy of the formulas obtained.