On Louchard's Asymptotic Series

Abstract

Recently G. Louchard obtained an asymptotic series $\sum_{j=0}^\infty\frac{I_j}{n^j}$ for the integral $\int_0^1[x^n+(1-x)^n]^{\frac1n}dx$ as $n\to\infty$, and computed $I_j$ for $j\le 5$ in terms of values of the Riemann zeta function. An interesting feature of the computation is that the $I_j$ are first obtained in terms of alternating multiple zeta values, but then everything except products of ordinary zeta values cancels out. We obtain similar formulas for $I_n$, $6\le n\le 9$, and conjecture a general formula for $I_n$ in terms of alternating multiple zeta values. We also conjecture that $I_n$ is a rational polynomial in the ordinary zeta values.