Algebraic Evaluations of Some Euler Integrals, Duplication Formulae for Appell's Hypergeometric Function F 1, and Brownian Variations

Ismail, Mourad E. H. ; Pitman, Jim

Canadian journal of mathematics, Tome 52 (2000) no. 5, pp. 961-981

Résumé

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 $n(n+1)/(2n+1)$ .

Détail

DOI : 10.4153/CJM-2000-040-3

Mots-clés : 33C65, 60J65, Brownian bridge, Gauss’s hypergeometric function, Lauricella’s multiple hypergeometric series, uniform order statistics, Appell functions

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