Sum formulas and duality theorems of multiple zeta values

Zeta Value Identities From Iterated Integrals With Additional Factors

“…Multiple zeta values with parameters were first introduced in (Chen, Chung & Eie, 2016;Eie & Lee, 2016) in order to provide a simple way to reprove the sum formula as well as the restricted sum formula (Eie, Liaw & Ong, 2009). Also they provided a systematic way to evaluate iterated integrals with parameters.…”

Identities for the Multiple Zeta (Star) Values

“…For some interesting results on generalized double zeta values (also called Euler sums), see [1,12]. The systematic study of multiple zeta values began in the early 1990s with the works of Hoffman [13], Zagier [26] and Borwein-Bradley-Broadhurst [2] and has continued with increasing attention in recent years (see [7,8,10,11]). The first systematic study of reductions up to depth 3 was carried out by Borwein and Girgensohn [6], where the authors proved that if p + q + r is even or less than or equal to 10 or p + q + r = 12, then triple zeta values ζ (q, p, r) (or ζ ⋆ (q, p, r)) can be expressed as a rational linear combination of products of zeta values and double zeta values.…”

Multiple zeta values and Euler sums

“…Many papers use the opposite convention, with the n i 's ordered by n 1 < n 2 < · · · < n k or n 1 ≤ n 2 ≤ · · · ≤ n k , see [11,12,14,15,21]. Multiple zeta values and multiple zeta star values were introduced and studied by Euler [17] in the old days.…”

“…From [5,11,12,[14][15][16], we know that multiple zeta values can be represented by iterated integrals (or Drinfeld integrals) over a simplex of weight dimension. Thus, we have the alternative (s 1 + s 2 + · · · + s k )-dimensional iterated-integral representation ζ (s 1 , s 2, · · · , s k ) = 1 0 Ω s 1 −1 w 1 Ω s 2 −1 w 2 · · · Ω s k −1 w k , s 1 > 1, (1.4) in which the integrand denotes a string of distinct differential 1-forms of type Ω := dx/x, and w j is given by A generalization of this duality formula can be found in [11,12,15,16]. On the other hand, the corresponding property of the duality formula for multiple zeta-star values was not known until recently.…”

“…be the set of natural numbers, and N 0 := N ∪ {0} be the set of positive integers and N \ {1} := {2, 3, 4, • • • }. For any multi-index S := (s 1 , s 2 , • • • , s k ) (s i ∈ N, k ∈ N, s 1 > 1), the general multiple zeta value ζ(S) and the multiple zeta star value ζ ⋆ (S) are defined, respectively, by convergent series ( [5,18,26,37]) Many papers use the opposite convention, with the n i 's ordered by [11,12,14,15,21]. Multiple zeta values and multiple zeta star values were introduced and studied by Euler [17] in the old days.…”