On Witten multiple zeta-functions associated with semisimple Lie algebras III

TL;DR

This paper establishes functional relations among Witten multiple zeta-functions linked to semisimple Lie algebras, enabling explicit evaluations at positive even integers using generalized Bernoulli numbers and introducing a generating function-based calculation method.

Contribution

It provides new functional relations for Witten zeta-functions tied to root systems and develops an algorithm for computing associated Bernoulli numbers.

Findings

Derived explicit formulas for Witten zeta-values at positive even integers.

Established symmetry-based functional relations among multivariable zeta-functions.

Introduced a generating function approach for calculating Bernoulli numbers of root systems.

Abstract

We prove certain general forms of functional relations among Witten multiple zeta-functions in several variables (or zeta-functions of root systems). The structural background of those functional relations is given by the symmetry with respect to Weyl groups. From those relations we can deduce explicit expressions of values of Witten zeta-functions at positive even integers, which is written in terms of generalized Bernoulli numbers of root systems. Furthermore we introduce generating functions of those Bernoulli numbers of root systems, by which we can give an algorithm of calculating Bernoulli numbers of root systems.