Proceedings of the Japan Academy, Series A, Mathematical Sciences

Zeta and $L$-functions and Bernoulli polynomials of root systems

Yasushi Komori, Kohji Matsumoto, and Hirofumi Tsumura

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Abstract

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.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 84, Number 5 (2008), 57-62.

Dates
First available in Project Euclid: 1 May 2008

Permanent link to this document
http://projecteuclid.org/euclid.pja/1209649653

Mathematical Reviews number (MathSciNet)
MR2415897

Digital Object Identifier
doi:10.3792/pjaa.84.57

Zentralblatt MATH identifier
1147.11053

Subjects
Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)

Keywords
Multiple zeta-function Witten zeta-function root systems simple Lie algeras analytic continuation functional relation

Citation

Komori, Yasushi; Matsumoto, Kohji; Tsumura, Hirofumi. Zeta and $L$-functions and Bernoulli polynomials of root systems. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 5, 57--62. doi:10.3792/pjaa.84.57. http://projecteuclid.org/euclid.pja/1209649653.


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