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Algebraic Evaluations of Some Euler Integrals, Duplication Formulae for Appell's Hypergeometric Function F1, and Brownian Variations

Published online by Cambridge University Press:  20 November 2018

Mourad E. H. Ismail  and
Jim Pitman
Mourad E. H. Ismail
Affiliation:
Department of Mathematics, University of South Florida, Tampa, FL 33620-5700, USA email: ismail@math.usf.edu
Jim Pitman
Affiliation:
Department of Statistics, University of California-Berkeley, Berkeley, CA 94720-3860, USA email: pitman@stat.berkeley.edu
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Abstract

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Explicit evaluations of the symmetric Euler integral $\int _{0}^{1}\,{{u}^{\alpha }}{{(1-u)}^{\alpha }}f(u)\,du$ are obtained for some particular functions $f$. These evaluations are related to duplication formulae for Appell’s hypergeometric function ${{F}_{1}}$ which give reductions of ${{F}_{1}}(\alpha ,\beta ,\beta ,2\alpha ,y,z)$ in terms of more elementary functions for arbitrary $\beta $ with $z=y/(y-1)$ and for $\beta =\alpha +\frac{1}{2}$ with arbitrary $y,z$. These duplication formulae generalize the evaluations of some symmetric Euler integrals implied by the following result: if a standard Brownian bridge is sampled at time 0, time 1, and at $n$ independent randomtimes with uniformdistribution on $[0,1]$, then the broken line approximation to the bridge obtained from these $n+2$ values has a total variation whose mean square is $n(n+1)/(2n+1)$.

Keywords

33C6560J65Brownian bridgeGauss’s hypergeometric functionLauricella’s multiple hypergeometric seriesuniform order statisticsAppell functions
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 5 , 01 October 2000 , pp. 961 - 981
Copyright
Copyright © Canadian Mathematical Society 2000

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