Another expression of the restricted sum formula of multiple zeta values

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.…”

“…In [22], M. Igarashi proved a generalization of the sum formula (see Proposition 3 in the reference [22]). 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…”