Open Access
May 2008 Zeta and $L$-functions and Bernoulli polynomials of root systems
Yasushi Komori, Kohji Matsumoto, Hirofumi Tsumura
Proc. Japan Acad. Ser. A Math. Sci. 84(5): 57-62 (May 2008). DOI: 10.3792/pjaa.84.57


This article is essentially an announcement of the papers [7,8,9,10] of the authors, though some of the examples are not included in those papers. We consider what is called zeta and $L$-functions of root systems which can be regarded as a multi-variable version of Witten multiple zeta and $L$-functions. Furthermore, corresponding to these functions, Bernoulli polynomials of root systems are defined. First we state several analytic properties, such as analytic continuation and location of singularities of these functions. Secondly we generalize the Bernoulli polynomials and give some expressions of values of zeta and $L$-functions of root systems in terms of these polynomials. Finally we give some functional relations among them by our previous method. These relations include the known formulas for their special values formulated by Zagier based on Witten’s work.


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Yasushi Komori. Kohji Matsumoto. Hirofumi Tsumura. "Zeta and $L$-functions and Bernoulli polynomials of root systems." Proc. Japan Acad. Ser. A Math. Sci. 84 (5) 57 - 62, May 2008.


Published: May 2008
First available in Project Euclid: 1 May 2008

zbMATH: 1147.11053
MathSciNet: MR2415897
Digital Object Identifier: 10.3792/pjaa.84.57

Primary: 11M41
Secondary: 40B05

Keywords: analytic continuation , functional relation , Multiple zeta-function , Root systems , simple Lie algeras , Witten zeta-function

Rights: Copyright © 2008 The Japan Academy

Vol.84 • No. 5 • May 2008
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