SIGMA 12 (2016), 101, 5 pages arXiv:1609.02827
Uniform Asymptotic Expansion for the Incomplete Beta Function
Gergő Nemes and Adri B. Olde Daalhuis
Maxwell Institute and School of Mathematics, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
Received September 12, 2016, in final form October 21, 2016; Published online October 25, 2016
In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 22.214.171.124] 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.
incomplete beta function; uniform asymptotic expansion.
pdf (259 kb)
tex (7 kb)
Johnson N.L., Kotz S., Balakrishnan N., Continuous univariate distributions, Vol. 2, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1995.
Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, Cambridge University Press, Cambridge, 2010, Release 1.0.13 of 2016-09-16, available at http://dlmf.nist.gov/.
Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996.
Temme N.M., Asymptotic methods for integrals, Series in Analysis, Vol. 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.