
Algebraic Evaluations of Some Euler Integrals, Duplication Formulae for Appell's Hypergeometric Function $F_1$, and Brownian Variations
Explicit evaluations of the symmetric Euler integral $\int_0^1
u^{\alpha} (1u)^{\alpha} f(u) \,du$ 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 $z = y/(y1)$ 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 $n(n+1)/(2n+1)$.
Keywords:Brownian bridge, Gauss's hypergeometric function, Lauricella's multiple hypergeometric series, uniform order statistics, Appell functions Categories:33C65, 60J65 