Explicit Formulas of Some Mixed Euler Sums via Alternating Multiple Zeta Values
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Explicit Formulas of Some Mixed Euler Sums via Alternating Multiple Zeta Values Ce Xu1,2,3 Received: 9 July 2019 / Revised: 3 January 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we present some new identities for (alternating) multiple zeta values and (alternating) Euler sums by using the method of iterated-integral representations of series. In particular, we prove five new evaluations of (alternating) mixed Euler sums via (alternating) multiple zeta values. Some interesting consequences and illustrative examples are considered. Keywords Multiple zeta (star) values · Alternating multiple zeta (star) values · Multiple harmonic (star) sums · Alternating multiple harmonic (star) sums · Euler sums Mathematics Subject Classification 11M06 · 11M32 · 40B05 · 33E20
1 Introduction The subjects of this paper are multiple zeta (star) values and Euler sums, which have been of interest to mathematicians and physicists for a long time. The multiple zeta (star) values and Euler sums have been useful in various areas of theoretical physics, such as the perturbative quantum field theory. Next, we give the detailed introductions of multiple zeta (star) values and Euler sums.
Communicated by Emrah Kilic.
B
Ce Xu [email protected] ; [email protected]
1
School of Mathematics and Statistics, Anhui Normal University, Wuhu 241000, People’s Republic of China
2
Multiple Zeta Research Center, Kyushu University, Motooka, Nishi-ku, Fukuoka 819-0389, Japan
3
School of Mathematical Sciences, Xiamen University, Xiamen 361005, People’s Republic of China
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C. Xu
Let s1 , . . . , sk be positive integers. The multiple harmonic sums (MHSs) and multiple harmonic star sums (MHSSs) are defined by ( [24,27])
ζn (s1 , s2 , . . . , sk ) :=
1 , · · · n skk
(1.1)
1 , · · · n skk
(1.2)
n s1 n s2 n≥n 1 >n 2 >···>n k ≥1 1 2
ζn (s1 , s2 , . . . , sk ) :=
n s1 n s2 n≥n 1 ≥n 2 ≥···≥n k ≥1 1 2
when n < k, then ζn (s1 , s2 , . . . , sk ) = 0, and ζn (∅) = ζn (∅) = 1. The integers k and w := s1 + · · · + sk are called the depth and the weight of a MH(S)S. For convenience, by s1 , . . . , s j d we denote the sequence of depth d j with d repetitions of s1 , . . . , s j . For example, ζn (2, 3, {1}2 ) = ζn (2, 3, 1, 1) , ζn (5, 2, {1}3 ) = ζn (5, 2, 1, 1, 1) . When taking the limit n → ∞ in (1.1) and (1.2), we get the multiple zeta values (MZVs, also called the Zagier sums) and the multiple zeta star values (MZSVs), respectively ( [7,8,35]): ζ (s1 , s2 , . . . , sk ) = lim ζn (s1 , s2 , . . . , sk ) , n→∞
ζ (s1 , s2 , . . . , sk ) =
lim ζ (s1 , s2 , . . . , sk ) n→∞ n
(1.3) (1.4)
defined for s2 , . . . , sk ≥ 1 and s1 ≥ 2 to ensure convergence of the series. From [8,11,14,15,17,18], we know that MZVs can be represented by iterated integrals (or Drinfeld integrals) over a simplex of weight dimension. Thus, we have the alternative (s1 + s2 + · · · + sk )-dimensional iterated-integral representation (see [8]) ζ s1 , s2, . . .
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