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

DOI:
https://doi.org/10.1090/S0025-5718-96-00729-6

MathSciNet review:
1344611

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Abstract | References | Similar Articles | Additional Information

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

Email:
doman@liv.ac.uk

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