Quasi-shuffle products revisited

Abstract

Quasi-shuffle products, introduced by the first author, have been useful in studying multiple zeta values and some of their analogues and generalizations. The second author, together with Kajikawa, Ohno, and Okuda, significantly extended the definition of quasi-shuffle algebras so it could be applied to multiple q-zeta values. This article extends some of the algebraic machinery of the first author's original paper to the more general definition, and demonstrates how various algebraic formulas in the quasi-shuffle algebra can be obtained in a transparent way. Some applications to multiple zeta values, interpolated multiple zeta values, multiple q-zeta values, and multiple polylogarithms are given.

Structure of Polyzetas and Explicit Representation on Transcendence Bases of Shuffle and Stuffle Algebras

“…The stuffle (or quasi-shuffle) product, i.e. the commutative product defined [22], for any w ∈ Y * , by…”

“…This means that the identity (49), which is equivalent to (13) [20,21], yields immediately the family of regularized double shuffle relations considered in [1,2,14,17,24,28] (see also [3,7,22,23,27]). …”

“…the commutative generating series of symbolic polyzetas, due to Ecalle [12] (Boutet de Monvel [3] and Racinet [14] had also given equivalent relations for the noncommutative generating series of symbolic polyzetas, see also [7]). Our method is quite different from these methods which produce linear relations and are based on the regularized double shuffle [2,22,24] and from identities among associators, due to Drinfel'd [9,10,16]. …”

Cofree Hopf algebras on Hopf bimodule algebras

“…Thanks to this isomorphism, we can concentrate our study on quantum quasi-shuffle algebras, which are the quantization of the classical quasishuffle algebras introduced by Newman and Radford [15], Hoffmann ( [7]), Guo and Keigher ( [6]) independently. They are constructed as a special case of quantum multibrace algebras ( [11]) and have some interesting properties and applications to other mathematical objects (cf.…”

The Double Shuffle Relations for Multiple eisenstein Series

“…To do this, we will reformulate the result of Hoffman [9,Theorem 2.5] in accordance with our situation.…”