Mathematics of Computation

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

and

. 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

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

1.: M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970.
2.: A. R. Didonato and A. H. Morris, Computation of the Incomplete Gamma Function Ratios and their Inverse, ACM Trans. Math. Software 12 (1986), 377-393.
3.: ------, Significant Digit Computation of the Incomplete Beta Function Ratios, ACM Trans. Math. Software 18 (1992), 360-373.
4.: B. G. S. Doman, C. J. Pursglove and W. M. Coen, A Set of Ada Packages for High Precision Calculations, ACM Trans. Math. Software 21 (1995), 416-431.
5.: J. L. Fields, A Note on the Asymptotic Expansion of the Ratio of Two Gamma Functions, Proc. Edinburgh Math. Soc. 15 (1966), 43-55. MR 34:379
6.: C. L. Frenzen, Error Bounds for Asymptotic Expansions of the Ratio of Two Gamma Functions, SIAM J. Math. Anal. 18 (1987), 890-896. MR 88d:33001
7.: Y. L. Luke, The Special Functions and their Approximations, Vol. I, Academic Press, New York, 1969. MR 39:3039
8.: E. C. Molina, Expansions for Laplacian Integrals in Terms of Incomplete Gamma Functions, International Congress of Mathematicians, Zurich, Bell System Technical Journal 11 (1932), 563-575 and Monograph B704.
9.: N. M. Temme, Incomplete Laplace Integrals: Uniform Asymptotic Expansion with Application to the Incomplete Beta Function, SIAM J. Math. Anal 18 (1987), 1638-1663. MR 89f:41036
10.: M. E. Wise, The use of the Negative Binomial Distribution in an Industrial Sampling Problem, Suppl. J. Roy. Statist. Soc. 8 (1946), 202-211. MR 9:49c
11.: ------, The Incomplete Beta Function as a Contour Integral and a Quickly Converging Series for its Inverse, Biometrika 37 (1950), 208-218. MR 12:724e

B. G. S. Doman
Affiliation: Department of Mathematical Sciences, University of Liverpool, PO Box 147, Liverpool L69 3BX, England
Email: doman@liv.ac.uk

DOI: 10.1090/S0025-5718-96-00729-6
PII: S 0025-5718(96)00729-6
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
Copyright of article: Copyright 1996, American Mathematical Society

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