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Proceedings of the London Mathematical Society Advance Access published online on July 20, 2009
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdp025
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© 2009 London Mathematical Society

On multiple Bernoulli polynomials and multiple L-functions of root systems

Yasushi Komori and Kohji Matsumoto

Graduate School of Mathematics
Nagoya University
Chikusa-ku
Nagoya 464-8602
Japan
kohjimat@math.nagoya-u.ac.jp

Hirofumi Tsumura

Department of Mathematics and Information Sciences
Tokyo Metropolitan University
1-1, Minami-Ohsawa
Hachioji
Tokyo 192-0397
Japan
tsumura@tmu.ac.jp

Received 17 April 2008. Revision received 17 March 2009.

We define generalized Bernoulli numbers, Bernoulli polynomials and multi-variable L-functions associated with root systems. We prove that the values of those L-functions at positive integers can be expressed in terms of those Bernoulli polynomials, and give an explicit formula for the latter. This result is a character analogue of Witten's volume formula for Witten's zeta-functions of semisimple Lie algebras. Furthermore, we show that the L-functions can be continued meromorphically to the whole space, and satisfy certain functional relations.

2000 Mathematics Subject Classification 11M41 (primary); 17B20, 40B05 (secondary).


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