Multiple $q$-Zeta Values

By David M. Bradley. 2004

150 citations recorded by the source corpus.

We introduce a q-analog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple q-zeta values satisfy a q-stuffle multiplica tion rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Additionally, multiple q-zeta values can be viewed as special values of the multiple q-polylogarithm, which admits a multiple Jackson q-integral representation whose limiting case is the Drinfel’d simplex integral for the ordinary multiple polylogarithm when q = 1. The multiple Jackson q-integral representation for multiple q-zeta values leads to a second multiplication rule satisfied by them, referred to as a q-shuffle. Despite this, it appears that many numerical relations satisfied by ordinary multiple zeta values have no interesting q-extension. For example, a suitable q-analog of Broadhurst’s formula for ζ( {3, 1} n ), if one exists, is likely to be rather complicated. Nevertheless, we show that a number of infinite classes of relations, inc luding Hoffman’s partition identities, Ohno’s cyclic sum identities, Granville’s sum formula, Euler’s convolution formula, Ohno’s generalized duality relation, and the derivation relations of Ihara and Kaneko extend to multiple q-zeta values.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/7ff46529056b92e40be536174d6434c72520397e

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Citations in the corpus

FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES (2005, methodology): “tier used to obtain his construction is readily seen to be the same as the one given by Ehrenborg [27] for the product of monomial quasi-symmetric functions and more recently the one given by Bradley [10] to explicitly describe stuﬄes and q-stuﬄes for multiple zeta values. Furthermore, the mixable s”
Mixable shuffles, quasi-shuffles and Hopf algebras (2005, background): “Hoffman’s quasi-shuffle product plays a prominent rôle in the recent studies of harmonic functions, quasi-symmetric functions, multiple zeta values [25, 27, 28, 4] (where in special cases it is also called stuffle product or harmonic product) and q-multiple zeta values [5].”
Quasi-shuffle products revisited (2016, methodology): “1 advantages: (i) it simultaneously applied to multiple zeta and multiple zetastar values and their extensions; and (ii) it could be applied to the q-series version of multiple zeta values studied in [4]. Here we show that some of the algebraic machinery developed in [7], particularly the coalgebra ”
Multiple q-zeta functions and multiple q-polylogarithms (2003, background): “Some of the ideas and results in this paper have been suggested or proved also by Bowman and Bradley [3] and Bradley [4] independently.”
Multiple zeta values vs. multiple zeta-star values (2011, methodology): “The formulas for MZVs, MZSVs, qMZVs and qMZSVs are established in [13,21,5] and [20], respectively.”
Renormalization of multiple zeta values (2006, background): “~a ′ ∗~b′ with the convention that ~a′ = 1 if m= 1, ~b′ = 1 if n= 1 and (a 1b1,~a ′ ∗~b′) = (a 1b1) if m= n= 1. Quasi-shuﬄe is also known as harmonic product [30] and coincides with the stuﬄe product [5, 7] in the study of MZVs. Variations of the stuﬄe product have also appeared in [10, 20]. See § 6”
Exts and Vertex Operators (2008, background): “ows at once from the following analog X n1≥n2>0 q−n2/2 [n2][n1]2 = X n>0 qn/2 [n]3 (12) of Euler’s ζ(2,1) = ζ(3) identity proven by Bradley. Namely, (12) is the k = n = 2 case of corollary 4 in [1]. One collects the terms in (9) to ﬁnish the proof of corollary 3. 2.3 Pieri-type formulas for Ja”
On relations for the multiple q-zeta values (2007, methodology): “In fact, the following equality holds up to q269: 4ζ[7, 2] + 6ζ[6, 3]− ζ[5, 4]− ζ[4, 5]− 6ζ[6, 2, 1]− 6ζ[6, 1, 2] −2ζ[5, 3, 1]− 7ζ[5, 2, 2]− 3ζ[5, 1, 3] + 2ζ[4, 4, 1]− ζ[4, 3, 2] +ζ[3, 5, 1] + ζ[3, 2, 4]− 3ζ[2, 5, 2] + 2ζ[5, 2, 1, 1] +2ζ[5, 1, 2, 1] + 2ζ[5, 1, 1, 2] + ζ[3, 3, 1, 2]− ζ[3, 2, 3, 1] −4”
Interpolation of multiple zeta and zeta-star values (2012, unknown): “gebra of h1 with respect to both products ∗ and ⋆, and Z: h0 ∗ −→ Rand Z⋆ : h0 ⋆ −→ Rare algebra homomorphisms. Another interesting example is the q-analogue of the multiple zeta values introduced in [B]. See [IKOO, Example 2] for details. 3. The interpolatingoperator and t-harmonic product 3.1. The”