Extended Double Shuffle Relations and the Generating Function of Triple Zeta Values of Any Fixed Weight

Abstract

Abstract. Extended double shuffle relations for multiple zeta values are obtained by using the fact that any product of regularized multiple zeta values has two different representations. In this paper, we give two formulas for the generating function of the triple zeta values of any fixed weight by the use of the extended double shuffle relations obtained as two-fold products of double and single zeta values and also as three-fold products of single zeta values. As applications of the formulas, we also obtain parameterized, weighted, and restricted sum formulas for triple zeta values.

New families of weighted sum formulas for multiple zeta values

“…Remark 2.2. It is conceivable that for every fixed positive integer d a compact formula of n−1 k=2 k d ζ(k, n − k) can be obtained by differentiating (11) repeatedly, similar to what we have done in Theorem 2.1. However, it seems to be a difficult problem to find a general formula for all d. Guo, Lei and the second author recently have made some progress along this direction, see [5].…”

“…The following corollary provides a sum formula relating some special type triple zeta and double zeta values. It also appeared in [11] as (5.12) which is a special case of [8,Thm. 2.3].…”

“…and proved some families of MZV identities. Machide [11] generalized this to depth three case using the extended (also called regularized) double shuffle relations. It is well-known that in order to get complete linear relations between MZVs one should consider regularized double shuffle relations.…”

On a variant of multiple zeta values of level two

“…We end this section by proposing the following conjecture as an analogue of Machide's formula [16,Corollary 4.1].…”

A study on multiple zeta values from the viewpoint of zeta-functions of root systems

“…Also Machide [26] gave certain restricted sum formulas for triple zeta values. Now recall our Corollaries 4.1 and 6.1.…”