The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

Bachmann, Henrik; Kuehn, Ulf

doi:10.48550/arxiv.1309.3920

Cited by 3 publications

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“…Theorem 2 asserts that the stuffle product (5) of the algebra MD reduces to the stuffle product of the algebra of MZVs in the limit as q → 1 − . This fact has been already established in [2].…”

Section: Theorem 2 ([2]mentioning

confidence: 60%

“…The first presence of the q-zeta brackets that are not reduced to ones from MD by the duality relation happens in weight 3. It is Z 2 2 and we find out that…”

Section: Reduction To Mono-bracketsmentioning

confidence: 82%

“…As explained in [2] (after the proof of Proposition 2.9), the product ⋄ is also associative. With the help of (4) define the stuffle product on the Q-algebra Q Z recursively by 1…”

Section: The Stuffle Productmentioning

confidence: 93%

“…In the other recent work [2,3] Bachmann and Kühn introduced and studied a different q-analogue of the MZVs, namely, [s 1 , . .…”

mentioning

confidence: 99%

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Multiple q-Zeta Brackets

Zudilin

2015

Mathematics

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Abstract:The multiple zeta values (MZVs) possess a rich algebraic structure of algebraic relations, which is conjecturally determined by two different (shuffle and stuffle) products of a certain algebra of noncommutative words. In a recent work, Bachmann constructed a q-analogue of the MZVs -the so-called bi-brackets -for which the two products are dual to each other, in a very natural way. We overview Bachmann's construction and discuss the radial asymptotics of the bi-brackets, its links to the MZVs, and related linear (in)dependence questions of the q-analogue.

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Section: Theorem 2 ([2]mentioning

confidence: 60%

“…The first presence of the q-zeta brackets that are not reduced to ones from MD by the duality relation happens in weight 3. It is Z 2 2 and we find out that…”

Section: Reduction To Mono-bracketsmentioning

confidence: 82%

“…As explained in [2] (after the proof of Proposition 2.9), the product ⋄ is also associative. With the help of (4) define the stuffle product on the Q-algebra Q Z recursively by 1…”

Section: The Stuffle Productmentioning

confidence: 93%

“…In the other recent work [2,3] Bachmann and Kühn introduced and studied a different q-analogue of the MZVs, namely, [s 1 , . .…”

mentioning

confidence: 99%

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Multiple q-Zeta Brackets

Zudilin

2015

Mathematics

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“…, h k }, h j := n 1 + n 2 + • • • + n j , and τ i (x) = x otherwise. Multiplying MZVs represented in either form, i.e., (2) or (3), results in Q-linear combinations of MZVs. Hence, the Q-vector space spanned by the real numbers (2) forms an algebra.…”

Section: Introductionmentioning

confidence: 99%

Unfolding the Double Shuffle Structure of -Multiple Zeta Values

Castillo-Medina¹,

Ebrahimi‐Fard²,

Manchon³

2015

Bull. Aust. Math. Soc.

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Multiple Divisor Functions and Multiple Zeta Values at Levle N

Yuan,

Zhao

2014

Preprint

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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 two-fold. First, we shall lay down the theoretical foundations of these values such as their regularizations and double shuffle relations. Second, we will generalize the multiple divisor functions (MDFs) defined by Bachman and Kühn to arbitrary level N and study their relations to MZVs at level N . These functions are all q-series and similar to MZVs, they have both weight and depth filtrations. But unlike that of MZVs, the product of MDFs usually has mixed weights; however, after projecting to the highest weight we can obtain an algebra homomorphism from MDFs to MZVs. Moreover, the image of the derivation D = q d dq on MDFs vanishes on the MZV side, which gives rise to many nontrivial Q-linear relations. In a sequel to this paper, we plan to investigate the nature of these relations.

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The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

Cited by 3 publications

References 0 publications

Multiple q-Zeta Brackets

Multiple q-Zeta Brackets

Unfolding the Double Shuffle Structure of -Multiple Zeta Values

Multiple Divisor Functions and Multiple Zeta Values at Levle N

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