An asymptotic expansion for the incomplete beta function

Abstract:

A new asymptotic expansion is derived for the incomplete beta function $I(a,b,x)$, which is suitable for large $a$, small $b$ and $x > 0.5$. This expansion is of the form \begin{equation*}I(a,b,x) \quad \sim \quad Q(b, -\gamma \log x) + {\frac {\Gamma (a + b)}{\Gamma (a) \Gamma (b)}} x^{\gamma } \sum ^{\infty }_{n=0}T_{n}(b,x)/ \gamma ^{n+1} , \end{equation*} where $Q$ is the incomplete Gamma function ratio and $\gamma = a + (b - 1)/2$ . This form has some advantages over previous asymptotic expansions in this region in which $T_{n}$ depends on $a$ as well as on $b$ and $x$.