# 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 $\int_0^1 u^{\alpha} (1-u)^{\alpha} f(u) du$ are obtained for some particular functions $f$. %such as $f(u) = [(1- yu)(1-zu)]^{\beta}$ for %$\beta = \alpha + \hf$ as well as for some other values. 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 for arbitrary $\beta$ with $z = y/(y-1)$ and for $\beta = \alpha + \hf$ 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 $n(n+1)/(2n+1)$. Key words and phrases: Brownian bridge, Gauss's hypergeometric function, Lauricella's multiple hypergeometric series, uniform order statistics, Appell functions.