The Algebra and Combinatorics of Shuffles andMultiple Zeta Values

Abstract

The algebraic and combinatorial theory of shuffles, introduced by Chen and Ree, is further developed and applied to the study of multiple zeta values. In particular, we establish evaluations for certain sums of cyclically generated multiple zeta values. The boundary case of our result reduces to a former conjecture of Zagier.

Multiple q-zeta values

“…where denotes the ordinary shuffle product [3,4,[6][7][8], a i = d q t/(t − α i ), b j = d q t/(t − β j ), i 0 = j 0 = 0, i c+1 = r + 1, j c+1 = s + 1, a r+1 , b s+1 = 1, and for all 1 i r and 1 j s,…”

“…The q-analog of a non-negative integer n is is the ordinary multiple zeta function [2][3][4][5][6][7][8][9]13,16]. In this paper, we make a detailed study of the multiple q-zeta function and its values at positive integer arguments.…”

A note on evaluations of multiple zeta values

“…In this article we provide a short and simple proof of Theorem 1 which refines the proof of Theorem 5.1 in [4].…”

Mixable shuffles, quasi-shuffles and Hopf algebras

“…DefineX = X ∪ {0}. Suppose that there is a pairing (4) [·, ·] :X ×X →X with the properties S0. [a, 0] = 0 for all a ∈X;…”