Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota–Baxter Algebras

Abstract

The multiple zeta values (MZVs) have been studied extensively in recent years. Currently there exist a few different types of q-analogs of the MZVs (q-MZVs) defined and studied by mathematicians and physicists. In this paper, we give a uniform treatment of these q-MZVs by considering their double shuffle relations (DBSFs) and duality relations. The main idea is a modification and generalization of the one used by Castillo Medina et al. who have considered the DBSFs of a special type of q-MZVs. We generalize their method to a few other types of q-MZVs including the one defined by the author in 2003. With different approach, Takeyama has already studied this type by "regularization" and observed that there exist Q-linear relations which are not consequences of the DBSFs. He also discovered a new family of relations which we call the duality relations in this paper. This deficiency of DBSFs occurs among other types, too, so we generalize the duality relations to all of these values and find that there are still some missing relations. This leads to the most general type of q-MZVs together with a new kind of relations called P-R relations which are used to lower the deficiencies further. As an application, we will confirm a conjecture of Okounkov on the dimensions of certain q-MZV spaces, either theoretically or numerically, for the weight up to 12. Some relevant numerical data are provided at the end.

A Dimension Conjecture for q-Analogues of Multiple Zeta Values

“…Usually a function f (q) is called a q-analogue of multiple zeta value, if lim q→1 f (q) is a multiple zeta value. There are various different models of q-analogues in the literature (See [27] for a nice overview). One of the first models was studied by Bradley [5] and Zhao [26] independently.…”

“…By a q-analogue of these numbers one usually understand q-series, which degenerate to multiple zeta values as q → 1. The algebraic structure of several models of q-analogues has been the subject of recent research (see [27] for an overview). Besides a conjecture of Okounkov in [17] for the dimension of the weight-graded spaces for a specific such model, no conjectures for the dimensions of the spaces of any of these q-analogues in a given weight and depth have occurred in the literature.…”

“…analogues of the stuffle and the shuffle product, is well understood (See for example [3], [9], [20], [21], [27]). The problem of understanding the dimension of the weight-graded spaces has been considered in [3], [4], [9], [17], [20], [21], [27]. On the other hand possible analogues of the Broadhurst-Kreimer conjecture for these q-analogues have not been proposed yet.…”

Multiple Eisenstein Series and q-Analogues of Multiple Zeta Values

“…The first works on this area are [Br], [Zh], [Sch] and [OOZ]. Possible double shuffle structures are discussed for example in [T], [EMM], [S] and [Zh2], where the last one gives also a nice overview of various different q-analogue models. Often these q-analogues have a product structure similar to the stuffle product of multiple zeta values.…”

Derivations on the algebra of multiple harmonic q-series and their applications

“…Here we recall their definitions and basic properties. See, e.g., [4,16] for details. Denote by z the C-submodule of H spanned by the set {e k } k∈ N .…”