Multiple $q$-Zeta Values

We give a new proof of the duality of multiple zeta values, which makes no use of the iterated integrals. The same method is also applicable to Ohno's relation for ($q$-)multiple zeta values.

Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves "coding" the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map \zeta: H^0 -> R, where H^0 is the graded rational vector space generated by the "admissible words" of the noncommutative polynomial algebra Q<...

The derivation relation is a well known relation among multiple zeta values, which was first obtained by Ihara, Kaneko and Zagier. The analogous formula for finite multiple zeta values, which we call the derivation relation for finite multiple zeta values, was conjectured by the third author and proved by the second author. In this paper, we show these two kinds of derivation relations are respectively equivalent to the Ohno type relations for multiple zeta values and finit...

We shall define the q-analogs of multiple zeta functions and multiple polylogarithms in this paper and study their properties, based on the work of Kaneko et al. and Schlesinger, respectively.

We prove some relations for the $q$-multiple zeta values ($q$MZV). They are $q$-analogues of the cyclic sum formula, the Ohno relation and the Ohno-Zagier relation for the multiple zeta values (MZV). We discuss the problem to determine the dimension of the space spanned by $q$MZV's over $\Q$, and present an application to MZV.

We explore the theory of multiple zeta values (MZVs) and some of their $q$-generalisations. Multiple zeta values are numerical quantities that satisfy several combinatorial relations over the rationals. These relations include two multiplicative relations, which arise naturally from comparison of the MZVs with an underlying algebraic structure. We generalise these concepts by introducing the parameter $q$ in such a way that as $q\to 1^-$ we return to the ordinary MZVs. Our sp...

We exhibit the double q-shuffle structure for the qMZVs recently introduced by Y. Ohno, J. Okuda and W. Zudilin.

Following Bachmann's recent work on bi-brackets and multiple Eisenstein series, Zudilin introduced the notion of multiple q-zeta brackets, which provides a q-analog of multiple zeta values possessing both shuffle as well as quasi-shuffle relations. The corresponding products are related in terms of duality. In this work we study Zudilin's duality construction in the context of classical multiple zeta values as well as various q-analogs of multiple zeta values. Regarding the f...

In this paper, we introduce the notion of generalized quasi-shuffle products and give a criterion for their associativity. These extend the quasi-shuffle products introduced by Hoffman, which are often used to describe the stuffle and shuffle product for multiple zeta values. For $q$-analogues of multiple zeta values, the description of an analogue for the shuffle product can often not be described with the classical notion of quasi-shuffle products. We show that our generali...

Multiple zeta values (MZVs) are generalizations of Riemann zeta values at positive integers to multiple variable setting. These values can be further generalized to level $N$ multiple polylog values by evaluating multiple polylogs at $N$-th roots of unity. In this paper, we consider another level $N$ generalization by restricting the indices in the iterated sums defining MZVs to congruences classes modulo $N$, which we call the MZVs at level $N$. The goals of this paper are t...