Algebraic evaluations of some Euler integrals, duplication formulae for Appell's hypergeometric function F1, and Brownian variations

Abstract

Explicit evaluations of the symmetric Euler integral ∫10 uα(1 - u)α f(u) du are obtained for some particular functions f. These evaluations are related to duplication formulae for Appell's hypergeometric function F1 which give reductions of F1 (α, β, β, 2α, y, z) in terms of more elementary functions for arbitrary β with z = y/(y - 1) and for β= α + 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 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 + l)/(2n + 1).