An asymptotic series for an integral

Abstract

We obtain an asymptotic series ∞ j=0 I j n j for the integral1 n dx as n → ∞, and compute I j in terms of alternating (or "colored") multiple zeta value. We also show that I j is a rational polynomial the ordinary zeta values, and give explicit formulas for j ≤ 12. As a byproduct, we obtain precise results about the convergence of norms of random variables and their moments. We study (U, 1 − U ) n as n tends to infinity and we also discuss (U 1 , U 2 , . . . , U r ) n for standard uniformly distributed random variables.

A note on harmonic number identities, Stirling series and multiple zeta values

“…First, we discuss expressions for ζ n ({1} k ). These expressions are, for example in terms of the Stirling numbers of the first kind, well-known (see for example Adamchik [1], Prodinger [21], Hoffman [15], Hoffman et al [17]), but perhaps not all parts of it and not in this notation.…”