Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 061, 15 pages      arXiv:1306.6164      https://doi.org/10.3842/SIGMA.2013.061
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

The Algebra of a q-Analogue of Multiple Harmonic Series

Yoshihiro Takeyama
Division of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan

Received June 27, 2013, in final form October 16, 2013; Published online October 22, 2013

Abstract
We introduce an algebra which describes the multiplication structure of a family of q-series containing a q-analogue of multiple zeta values. The double shuffle relations are formulated in our framework. They contain a q-analogue of Hoffman's identity for multiple zeta values. We also discuss the dimension of the space spanned by the linear relations realized in our algebra.

Key words: multiple harmonic series; q-analogue.

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References

  1. Bradley D.M., Multiple q-zeta values, J. Algebra 283 (2005), 752-798, math.QA/0402093.
  2. Drinfel'd V.G., On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q−/Q), Leningrad Math. J. 2 (1991), 829-860.
  3. Hoffman M.E., Multiple harmonic series, Pacific J. Math. 152 (1992), 275-290.
  4. Hoffman M.E., The algebra of multiple harmonic series, J. Algebra 194 (1997), 477-495.
  5. Hoffman M.E., Ohno Y., Relations of multiple zeta values and their algebraic expression, J. Algebra 262 (2003), 332-347, math.QA/0010140.
  6. Ihara K., Kaneko M., Zagier D., Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006), 307-338.
  7. Kaneko M., Kurokawa N., Wakayama M., A variation of Euler's approach to values of the Riemann zeta function, Kyushu J. Math. 57 (2003), 175-192, math.QA/0206171.
  8. Ohno Y., Okuda J., Zudilin W., Cyclic q-MZSV sum, J. Number Theory 132 (2012), 144-155.
  9. Okuda J., Takeyama Y., On relations for the multiple q-zeta values, Ramanujan J. 14 (2007), 379-387, math.QA/0402152.
  10. Zagier D., Values of zeta functions and their applications, in First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math., Vol. 120, Birkhäuser, Basel, 1994, 497-512.
  11. Zhao J., Multiple q-zeta functions and multiple q-polylogarithms, Ramanujan J. 14 (2007), 189-221.

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