[Paper Review] Some relations deduced from regularized double shuffle relations of multiple zeta values

Zhonghua Li, Chen Qin|arXiv (Cornell University)|Oct 18, 2016

Advanced Mathematical Identities39 references3 citations

TL;DR

This paper derives multiple algebraic relations among multiple zeta values (MZVs) exclusively from the regularized double shuffle relations, which are conjectured to generate all such relations. It establishes key formulas including the weighted sum formula of Guo and Xie, evaluation formulas for MZVs with even arguments, and generalized restricted sum formulas, providing strong evidence that these relations are sufficient to derive known identities in the MZV space.

ABSTRACT

It is conjectured that the regularized double shuffle relations give all algebraic relations among the multiple zeta values, and hence all other algebraic relations should be deduced from the regularized double shuffle relations. In this paper, we provide as many as the relations which can be derived from the regularized double shuffle relations, for example, the weighted sum formula of L. Guo and B. Xie, some evaluation formulas with even arguments and the restricted sum formulas of M. E. Hoffman and their generalizations.

Motivation & Objective

• To investigate whether all algebraic relations among multiple zeta values can be deduced from the regularized double shuffle relations.
• To derive known identities—such as the weighted sum formula and restricted sum formulas—using only the regularized double shuffle framework.
• To provide evidence that the regularized double shuffle relations are sufficient to generate major classes of MZV identities.
• To explore connections between regularized double shuffle relations and other conjectural systems of relations, such as associator and Kawashima relations.

Proposed method

• The authors work within the algebraic framework of the double shuffle relations, focusing on their regularized version to handle divergent series.
• They use stuffle (stuffle) products and iterated integral representations to derive algebraic identities among MZVs.
• The method involves summing over indices with fixed weight and applying the regularized double shuffle relations to derive linear combinations of MZVs.
• They analyze the coefficients in the regularized relations and prove their equivalence to known formulas using binomial identities and generating functions.
• The proof relies on a $Π$-algebraic structure and the use of a $Π$-linear map $Z_R$ to map indices to real numbers while preserving relations.
• They verify conjectural identities by summing over symmetric index families and showing equality under the stuffle product.

Experimental results

Research questions

• RQ1 Can the weighted sum formula of Guo and Xie be derived solely from the regularized double shuffle relations?
• RQ2 Are the restricted sum formulas of Hoffman and their generalizations consequences of the regularized double shuffle relations?
• RQ3 Can evaluation formulas for MZVs with even arguments be deduced from the regularized double shuffle framework?
• RQ4 What is the relationship between the regularized double shuffle relations and other conjectural systems like associator or Kawashima relations?
• RQ5 Do the Brown-Zagier and Ohno-Zagier relations follow from the regularized double shuffle relations?

Key findings

• The weighted sum formula of L. Guo and B. Xie is derived as a consequence of the regularized double shuffle relations.
• The restricted sum formula of M. E. Hoffman and its generalizations are shown to be derivable from the regularized double shuffle framework.
• Evaluation formulas for multiple zeta values with even arguments are deduced using the regularized double shuffle relations.
• The Ohno-Zagier relation and its corollaries are derived from the regularized double shuffle relations, confirming their consistency with the conjecture.
• The conjectural Brown-Zagier relation is supported by evidence derived from the regularized double shuffle framework.
• The coefficients in the regularized double shuffle relations are computed explicitly, and their sum over symmetric index families leads to known identities, confirming the method's validity.