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The Quarterly Journal of Mathematics Advance Access originally published online on December 11, 2007
The Quarterly Journal of Mathematics 2008 59(1):55-83; doi:10.1093/qmath/ham025
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

A NEW METHOD OF PRODUCING FUNCTIONAL RELATIONS AMONG MULTIPLE ZETA-FUNCTIONS

Kohji Matsumoto {dagger}

Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

Hirofumi Tsumura{ddagger}

Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan

{ddagger} Corresponding author. E-mail: tsumura{at}tmu.ac.jp

Received 2 March 2007; revised 23 April 2007
  Abstract

In this paper, we introduce a new method of producing functional relations among multiple zeta-functions. This method can be regarded as a kind of multiple analogue of Hardy's one of proving the functional equation for the Riemann zeta-function. Using this method, we give new functional relations for multiple zeta-functions. In particular, substituting positive integers into variables of them, we obtain known relation formulas for the multiple zeta-values. Furthermore, applying our method to a certain series involving hyperbolic sine functions, we can obtain certain multiple analogues of the known results given by Cauchy, Ramanujan, Berndt and so on.

{dagger} E-mail: kohjimat{at}math.nagoya-u.ac.jp


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