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A FAMILY OF MULTIPLE HARMONIC SUM AND MULTIPLE ZETA STAR VALUE IDENTITIES

Part of: Zeta and $L$-functions: analytic theory Sequences and sets

Published online by Cambridge University Press:  28 May 2014

Erin Linebarger  and
Jianqiang Zhao
Erin Linebarger
Affiliation:
Department of Mathematics, Eckerd College, St. Petersburg, FL 33711, U.S.A. email emlineba@eckerd.edu
Jianqiang Zhao
Affiliation:
Kavli Institute for Theoretical Physics China, Beijing, China Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Department of Mathematics, Eckerd College, St. Petersburg, FL 33711, U.S.A. email zhaoj@eckerd.edu
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Abstract

In this paper we present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehrood et al  [Trans. Amer. Math. Soc. (to appear)]. We then apply it to obtain a family of identities relating multiple zeta star values to alternating Euler sums. In such a typical identity the entries of the multiple zeta star values consist of blocks of arbitrarily long 2-strings separated by positive integers greater than two while the largest depth of the alternating Euler sums depends only on the number of 2-string blocks but not on their lengths.

MSC classification

Primary: 11B65: Binomial coefficients; factorials; $q$-identities 11M32: Multiple Dirichlet series and zeta functions and multizeta values

Information

Type
Research Article
Information
Mathematika , Volume 61 , Issue 1 , January 2015 , pp. 63 - 71
Copyright
Copyright © University College London 2014 

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References

Broadhurst, D. J., Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams. Preprint, 1996, arXiv:hep-th/9612012.Google Scholar
Brown, F., Mixed Tate motives over Spec(ℤ). Ann. of Math. (2) 175 2012, 949–976.Google Scholar
Deligne, P., Le groupe fondamental de la Gm−𝜇N, pour N = 2, 3, 4, 6 ou 8. Publ. Math. Inst. Hautes Études Sci. 112 2010, 101–141.Google Scholar
Deligne, P. and Goncharov, A., Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. Éc. Norm. Supér. (4) 38(1) 2005, 1–56.CrossRefGoogle Scholar
Goncharov, A. B. and Manin, Y. I., Multiple 𝜁-motives and moduli spaces M 0, n. Compositio Math. 140 2004, 1–14.CrossRefGoogle Scholar
Hessami Pilehrood, Kh. and Hessami Pilehrood, T., On $q$-analogues of two-one formulas for multiple harmonic sums and multiple zeta star values. Preprint, 2013, arXiv:1304.0269 [NT].CrossRefGoogle Scholar
Hessami Pilehrood, Kh., Hessami Pilehrood, T. and Tauraso, R., New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner’s series. Trans. Amer. Math. Soc. (to appear), doi: http://dx.doi.org/10.1090/S0002-9947-2013-05980-6.CrossRefGoogle Scholar
Hessami Pilehrood, Kh., Hessami Pilehrood, T. and Zhao, J., On $q$-analogues of some families of multiple harmonic sum and multiple zeta star value identities. Preprint, 2013, arXiv:1307.7985.Google Scholar
Zagier, D., Evaluation of the multiple zeta values 𝜁(2, …, 2, 3, 2, …, 2). Ann. of Math. (2) 175 2012, 977–1000.CrossRefGoogle Scholar
Zhao, J., Wolstenholme type Theorem for multiple harmonic sums. Int. J. Number Theory 4(1) 2008, 73–106.Google Scholar
Zhao, J., Identity families of multiple harmonic sums and multiple zeta (star) values. Preprint, 2013, arXiv:1303.2227.Google Scholar