An asymptotic series for an integral

Abstract

We obtain an asymptotic series $$\sum _{j=0}^\infty \frac{I_j}{n^j}$$ for the integral $$\int _0^1[x^n+(1-x)^n]^{\frac{1}{n}}\mathrm{{d}}x$$ as $$n\rightarrow \infty $$ , and compute $$I_j$$ in terms of alternating (or “colored”) multiple zeta values. We also show that $$I_j$$ is a rational polynomial in the ordinary zeta values, and give explicit formulas for $$j\le 12$$ . As a by-product, we obtain precise results about the convergence of norms of random variables and their moments. We study $$Z_n=\Vert (U,1-U)\Vert _n$$ as n tends to infinity and we also discuss $$W_n=\Vert (U_1,U_2,\dots ,U_r)\Vert _n$$ for standard uniformly distributed random variables.