Multiple q-Zeta Brackets

Zudilin, Wadim

doi:10.3390/math3010119

Cited by 12 publications

(9 citation statements)

References 10 publications

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“…Also interesting is Bachmann's model since it gives a deep connection to modular forms that play an important role in the theory of MZVs as already Gangl, Kaneko, and Zagier (in [GKZ]) and Broadhurst and Kreimer (in [BK]) have shown. For more details about the various models, we refer to the original works [Bra], [Zh1], [Sch], [Zu2], [Ba2], [Tak], [OOZ], [Oko], such as to [Zh2], where the author gives an overview of the models and their history.…”

Section: Models Of Qmzvsmentioning

confidence: 99%

“…Bi-brackets and their structure are well known, for more details than in this section we refer to [Ba3], [Ba4], [Ba5], [BK1], [BK2], [Zu2].…”

Section: Bi-bracketsmentioning

confidence: 99%

“…Remark 2.45. Zudilin's model of qMZVs, multiple q-zeta brackets, is closely related to bi-brackets ([Zu2] for details). They are defined for k 1 , .…”

Section: Bi-bracketsmentioning

confidence: 99%

“…We consider this model not in more detail since it is "more or less" Bachmann's model of bibrackets, and the connection to bi-brackets is also studied very well in [Zu2].…”

Section: Bi-bracketsmentioning

confidence: 99%

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A unified approach to qMZVs

Brindle¹

2021

Preprint

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Section: Models Of Qmzvsmentioning

confidence: 99%

“…Bi-brackets and their structure are well known, for more details than in this section we refer to [Ba3], [Ba4], [Ba5], [BK1], [BK2], [Zu2].…”

Section: Bi-bracketsmentioning

confidence: 99%

“…Remark 2.45. Zudilin's model of qMZVs, multiple q-zeta brackets, is closely related to bi-brackets ([Zu2] for details). They are defined for k 1 , .…”

Section: Bi-bracketsmentioning

confidence: 99%

“…We consider this model not in more detail since it is "more or less" Bachmann's model of bibrackets, and the connection to bi-brackets is also studied very well in [Zu2].…”

Section: Bi-bracketsmentioning

confidence: 99%

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A unified approach to qMZVs

Brindle¹

2021

Preprint

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“…See [28] for details. Inspired by Bachmann's intriguing work [1], Zudilin presents in [30] a particular model called multiple q-zeta brackets, which possesses a natural quasi-shuffle product. After multiplying Zudilin's multiple q-zeta brackets with a certain positive integer power of 1´q one obtains ordinary MZVs in the classical limit q Ñ 1.…”

Section: Introductionmentioning

confidence: 99%

Duality and (q-)multiple zeta values

Ebrahimi‐Fard

Manchon

Singer

2016

Advances in Mathematics

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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 former we identify the derivation relation of order two with a Hoffman-Ohno type relation. Then we describe relations between the Ohno-Okuda-Zudilin q-multiple zeta values and the Schlesinger-Zudilin q-multiple zeta values.

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

Zhao¹

2020

Springer Proceedings in Mathematics &Amp; Statistics

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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.

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Multiple q-Zeta Brackets

Cited by 12 publications

References 10 publications

A unified approach to qMZVs

A unified approach to qMZVs

Duality and (q-)multiple zeta values

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

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