An asymptotic expansion for the incomplete beta function

Abstract

A new asymptotic expansion is derived for the incomplete beta function I ( a , b , x ) I(a,b,x) , which is suitable for large a a , small b b and x > 0.5 x > 0.5 . This expansion is of the form I ( a , b , x ) ∼ Q ( b , − γ log ⁡ x ) + Γ ( a + b ) Γ ( a ) Γ ( b ) x γ ∑ n = 0 ∞ T n ( b , x ) / γ n + 1 , \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 Q is the incomplete Gamma function ratio and γ = a + ( b − 1 ) / 2 \gamma = a + (b - 1)/2 . This form has some advantages over previous asymptotic expansions in this region in which T n T_{n} depends on a a as well as on b b and x x .