The algebra of bi-brackets and regularized multiple Eisenstein series

Bachmann, Henrik

doi:10.1016/j.jnt.2018.12.006

Cited by 15 publications

(37 citation statements)

References 11 publications

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“…We have similar tables for various subspaces like the positive bi-brackets Although, our experiments support conjectures (B1) and (B3), we were not able to prove the weaker claims that the positive bi-brackets respectively the 123-brackets generate sub-algebras of Z q . In [1] the conjecture (B2) was stated the first time and therein examples which complement those in [9] (Theorem 5.5) were given. We will give a proof of this theorem at the end of this section.…”

Section: Computational Evidences For the Conjecturesmentioning

confidence: 93%

“…The expected equality in Proposition 2.5 would hold if the brackets [2], [4], [6] and the odd brackets [1], [3], .. together with all of their derivatives were algebraically independent, but by now only partial results for linear independence are available ( [24], [28]).…”

Section: Bi-brackets and Quasi-modular Formsmentioning

confidence: 99%

“…We refer to these q-series as bi-brackets of depth l and weight s 1 + · · · + s l + r 1 + · · · + r l . They were introduced by the first author in [1] and their algebraic structure is wellunderstood and described in the papers [1], [2], [3], [30]. The bi-brackets have a natural connection to quasi-modular forms (for SL 2 (Z)), since for even k the Fourier expansion of the classical Eisenstein series G k of weight k is given by k 0 plus an appropriate constant term.…”

Section: Introductionmentioning

confidence: 99%

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A Dimension Conjecture for q-Analogues of Multiple Zeta Values

Bachmann

Kuehn

2020

Springer Proceedings in Mathematics &Amp; Statistics

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We study a class of q-analogues of multiple zeta values given by certain formal q-series with rational coefficients. After introducing a notion of weight and depth for these q-analogues of multiple zeta values we present dimension conjectures for the spaces of their weight-and depth-graded parts, which have a similar shape as the conjectures of Zagier and Broadhurst-Kreimer for multiple zeta values. n 1 >···>n l >0 1 n s 1 1 . . . n s l l .By s 1 + · · · + s l we denote its weight, by l its depth and we write Z for the Q-vector space spanned by all MZVs. It is a well-known fact that the space Z is a Q-algebra and that there are two different ways, known as respectively the stuffle and shuffle product formulas, which express the product of two MZVs as a Q-linear combination of MZVs.

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Section: Computational Evidences For the Conjecturesmentioning

confidence: 93%

Section: Bi-brackets and Quasi-modular Formsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Dimension Conjecture for q-Analogues of Multiple Zeta Values

Bachmann

Kuehn

2020

Springer Proceedings in Mathematics &Amp; Statistics

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“…We study a specific connection of multiple zeta values and modular forms given by multiple Eisenstein series. This work is an example driven overview article and summary of the results obtained in the works [BK], [BT], [Ba2] and [BK2].…”

Section: Introductionmentioning

confidence: 99%

“…. , s l ] * version of the ordinary brackets as certain linear combination of bi-brackets as introduced in [Ba2]. These objects fulfill the same shuffle and stuffle products as multiple zeta values respectively.…”

Section: Introductionmentioning

confidence: 99%

Multiple Eisenstein Series and q-Analogues of Multiple Zeta Values

Bachmann

2020

Springer Proceedings in Mathematics &Amp; Statistics

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Partitions, multiple zeta values and the q-bracket

Bachmann,

van Ittersum

2023

Sel. Math. New Ser.

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We provide a framework for relating certain q-series defined by sums over partitions to multiple zeta values. In particular, we introduce a space of polynomial functions on partitions for which the associated q-series are q-analogues of multiple zeta values. By explicitly describing the (regularized) multiple zeta values one obtains as $$q\rightarrow 1$$ q → 1 , we extend previous results known in this area. Using this together with the fact that other families of functions on partitions, such as shifted symmetric functions, are elements in our space will then give relations among (q-analogues of) multiple zeta values. Conversely, we will show that relations among multiple zeta values can be ‘lifted’ to the world of functions on partitions, which provides new examples of functions for which the associated q-series are quasimodular.

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The algebra of bi-brackets and regularized multiple Eisenstein series

Cited by 15 publications

References 11 publications

A Dimension Conjecture for q-Analogues of Multiple Zeta Values

A Dimension Conjecture for q-Analogues of Multiple Zeta Values

Multiple Eisenstein Series and q-Analogues of Multiple Zeta Values

Partitions, multiple zeta values and the q-bracket

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