An asymptotic series for an integral
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An asymptotic series for an integral Michael E. Hoffman1 · Markus Kuba2
· Moti Levy3 · Guy Louchard4
Received: 12 March 2018 / Accepted: 1 December 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 1 n Ij n n1 We obtain an asymptotic series ∞ j=0 n j for the integral 0 [x + (1 − x) ] dx as n → ∞, 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 ≤ 12. As a by-product, we obtain precise results about the convergence of norms of random variables and their moments. We study Z n = (U , 1 − U )n as n tends to infinity and we also discuss Wn = (U1 , U2 , . . . , Ur )n for standard uniformly distributed random variables. Keywords Multiple zeta values · Alternating multiple zeta values · Asymptotic expansion · Norms of random variables Mathematics Subject Classification 11M32 · 60C05
B
Markus Kuba [email protected] Michael E. Hoffman [email protected] Moti Levy [email protected] Guy Louchard [email protected]
1
Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402, USA
2
Department of Applied Mathematics and Physics, Fachhochschule Technikum Wien, Höchstädtpl. 6, 1200 Wien, Austria
3
Technion - Israel Institute of Technology, Rehovot, Israel
4
Faculté des sciences, Université libre de Bruxelles, CP212, Boulevard du Triomphe 2, 1050 Bruxelles, Belgium
123
M. E. Hoffman et al.
1 Introduction Let I (n) =
1
1
[x n + (1 − x)n ] n dx.
(1)
0
We shall obtain an asymptotic series as n → ∞, I (n) = I0 +
I1 I3 I2 + 2 + 3 + ··· n n n
The problem of calculating I0 and I2 was originally introduced as a problem in the American Mathematical Monthly [8] in 2016. It turned out that I0 = 34 and I2 is given
by a logarithmic integral. The value of I2 is given by 18 ζ (2) = π48 . The integral I (n) has been discussed further in [14], together with a different problem proposed by M. D. Ward, in order to shed more light on the appearance of (multiple) zeta values in the asymptotic expansion of I (n). Therein, it is treated by a different approach using analytic combinatorics, Euler sums and polylogarithms, leading to the first few terms I0 to I7 in terms of multiple zeta values. Here, we give a complete expansion of I (n). The coefficients Ik can be written in terms of alternating or “colored” multiple zeta values. The multiple zeta values are defined by 2
ζ (i 1 , . . . , i k ) =
i1 n 1 >···>n k ≥1 n 1
1 · · · n ikk
for positive integers i 1 , . . . , i k with i 1 > 1. This notation can be extended to alternating or “colored” multiple zeta values by putting a bar over those exponents with an associated sign in the numerator, as in ¯ 1, ¯ 1) = ζ (3,
n 1 >n 2 >n 3
(−1)n 1 +n 2 . n 31 n 2 n 3 ≥1
¯ = − log 2 Note that ζ (i 1 , i 2 , . . . , i k ) converges unless i 1 is an unbarred 1. We have ζ (1) and ζ (n) ¯ = (21−n − 1)ζ (n) for n ≥ 2. Alternating multiple zeta values have been extensively studied, and some identitie
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