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**Zeta and \(L\)-functions and Bernoulli polynomials of root systems.**
*(English)*
Zbl 1147.11053

Summary: This article is essentially an announcement of the papers [Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras, II, III, IV (preprint); On multiple Bernoulli polynomials and multiple \(L\)-functions of root systems (preprint)] 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 D. Zagier [Prog. Math. 120, 497–512 (1994; Zbl 0822.11001)] based on Witten’s work.

### MSC:

11M41 | Other Dirichlet series and zeta functions |

17B20 | Simple, semisimple, reductive (super)algebras |

40B05 | Multiple sequences and series |

11G55 | Polylogarithms and relations with \(K\)-theory |

11B68 | Bernoulli and Euler numbers and polynomials |

### Keywords:

multiple zeta-function; Witten zeta-function; root systems; simple Lie algebras; analytic continuation; functional relation### Citations:

Zbl 0822.11001
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\textit{Y. Komori} et al., Proc. Japan Acad., Ser. A 84, No. 5, 57--62 (2008; Zbl 1147.11053)

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