Depth Reduction of a Class of Witten Zeta Functions

Abstract

The first and third authors are supported by the National Natural Science Foundation of China, Project 10871169.We show that if a, b, c, d, f are positive integers such that a + b + c + d + f is even, then the Witten zeta value ζ sl (4) (a, b, c, d, 0, f ) is expressible in terms of Witten zeta functions with fewer arguments.

Functional Relations for Zeta-Functions of Root Systems

“…By the same technique, Nakamura [35] obtained functional relations among zeta-functions of B 2 , A 3 , and of A 2 with characters. Yet another (rather elementary) method was proposed by Zhou, Bradley and Cai [43], who stated certain functional relations among zeta-functions of A 3 . Inspired by [43], Ikeda and Matsuoka [8] gave functional relations among zeta-functions of A 2 , A 3 and A 4 (with correcting some inaccurate point in [43]).…”

“…The result in this section is given by considering partial fraction decompositions and partial summations of harmonic sums, and the method here is totally different from that stated in the preceding sections. The main technique is similar to that used in [39] (and methodologically some common feature with [8,43], though they studied the case of type A r ). Let…”

Zeta-functions of root systems and Poincaré polynomials of Weyl groups

“…The first example is a functional relation between ζ(s) and ζ 2 ((s 1 , s 2 , s 3 ), 0; A 2 ) discovered by the third-named author [26], which interpolates certain value relations among ζ(k) and ζ EZ,2 (k 1 , k 2 ) (k, k 1 , k 2 ∈ N). After this discovery, various other functional relations among zeta-functions of root systems have been reported (the aforementioned papers of the authors, Nakamura [20] [21], Zhou et al [29], Onodera [22], and Ikeda and Matsuoka [3]).…”