×

Uniform asymptotic expansion for the incomplete beta function. (English) Zbl 1351.41026

Summary: In [N. M. Temme, Special functions. An introduction to the classical functions of mathematical physics. New York, NY: Wiley (1996; Zbl 0856.33001), Section 11.3.3.1] a uniform asymptotic expansion for the incomplete beta function was derived. It was not obvious from those results that the expansion is actually an asymptotic expansion. We derive a remainder estimate that clearly shows that the result indeed has an asymptotic property, and we also give a recurrence relation for the coefficients.

MSC:

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)

Citations:

Zbl 0856.33001
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] N{IST} handbook of mathematical functions, xvi+951, (2010), U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, Cambridge University Press, Cambridge · Zbl 1198.00002
[2] Johnson, Norman L. and Kotz, Samuel and Balakrishnan, N., Continuous univariate distributions, {V}ol. 2, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, xxii+719, (1995), John Wiley & Sons, Inc., New York · Zbl 0821.62001
[3] Temme, Nico M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, xiv+374, (1996), John Wiley & Sons, Inc., New York · Zbl 0856.33001 · doi:10.1002/9781118032572
[4] Temme, Nico M., Asymptotic methods for integrals, Series in Analysis, 6, xxii+605, (2015), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ · Zbl 1312.41002 · doi:10.1142/9195
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.