
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 qAnalogue of Multiple Harmonic Series
Yoshihiro Takeyama
Division of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 3058571, 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 qseries
containing a qanalogue of multiple zeta values.
The double shuffle relations are formulated in our framework.
They contain a qanalogue 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; qanalogue.
pdf (383 kb)
tex (16 kb)
References
 Bradley D.M., Multiple qzeta values, J. Algebra 283
(2005), 752798, math.QA/0402093.
 Drinfel'd V.G., On quasitriangular quasiHopf algebras and on a group that is
closely connected with Gal(Q^{−}/Q),
Leningrad Math. J. 2 (1991), 829860.
 Hoffman M.E., Multiple harmonic series, Pacific J. Math. 152
(1992), 275290.
 Hoffman M.E., The algebra of multiple harmonic series, J. Algebra
194 (1997), 477495.
 Hoffman M.E., Ohno Y., Relations of multiple zeta values and their algebraic
expression, J. Algebra 262 (2003), 332347,
math.QA/0010140.
 Ihara K., Kaneko M., Zagier D., Derivation and double shuffle relations for
multiple zeta values, Compos. Math. 142 (2006), 307338.
 Kaneko M., Kurokawa N., Wakayama M., A variation of Euler's approach to
values of the Riemann zeta function, Kyushu J. Math. 57
(2003), 175192, math.QA/0206171.
 Ohno Y., Okuda J., Zudilin W., Cyclic qMZSV sum, J. Number
Theory 132 (2012), 144155.
 Okuda J., Takeyama Y., On relations for the multiple qzeta values,
Ramanujan J. 14 (2007), 379387, math.QA/0402152.
 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, 497512.
 Zhao J., Multiple qzeta functions and multiple qpolylogarithms,
Ramanujan J. 14 (2007), 189221.

