Uniform Asymptotic Expansion for the Incomplete Beta Function

TL;DR

This paper rigorously establishes the asymptotic nature of a uniform expansion for the incomplete beta function, providing a remainder estimate and recurrence relations for coefficients to enhance mathematical understanding.

Contribution

It proves the asymptotic property of a previously derived uniform expansion for the incomplete beta function and supplies recurrence relations for its coefficients.

Findings

Remainder estimate confirming asymptotic behavior

Recurrence relations for expansion coefficients

Enhanced understanding of the incomplete beta function asymptotics

Abstract

In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the incomplete beta function was derived. It was not obvious from those results that the expansion is actually an asymptotic expansion. We derive a remainder estimate that clearly shows that the result indeed has an asymptotic property, and we also give a recurrence relation for the coefficients.