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An asymptotic expansion for the incomplete beta function

Author: B. G. S. Doman
Journal: Math. Comp. 65 (1996), 1283-1288
MSC (1991): Primary 33B20; Secondary 65D20
MathSciNet review: 1344611
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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 $x$.

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Additional Information

B. G. S. Doman
Affiliation: Department of Mathematical Sciences, University of Liverpool, PO Box 147, Liverpool L69 3BX, England

Keywords: Gamma function ratio, incomplete Beta function, Chi-square distribution, Student’s distribution, $F$ distribution
Received by editor(s): March 16, 1995
Received by editor(s) in revised form: June 26, 1995
Article copyright: © Copyright 1996 American Mathematical Society