Algebraic Evaluations of Some Euler Integrals, Duplication Formulae for Appell's Hypergeometric Function $F_1$, and Brownian Variations

http://dx.doi.org/10.4153/CJM-2000-040-3
Canad. J. Math. 52(2000), 961-981

  Published:2000-10-01
 Printed: Oct 2000

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 $\beta = \alpha + \half$ 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 random times with uniform distribution 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 $n(n+1)/(2n+1)$.