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

On multiple higher Mahler measures and Witten zeta values associated with semisimple Lie algebras

“…Some explicit formulas for ζ W (2k, g) (k ∈ N) were given by Mordell [9], Zagier [16], Subbarao and Sitaramachandrarao [12] and Gunnells and Sczech [1]. Further Matsumoto "CNTP-6-4-A2-SASAKI" -2013/6/3 -12:58 -page 773 -#3 and Tsumura [8] and Komori et al [3] introduced the multi-variable Witten zeta-functions associated with semisimple Lie algebras, and evaluated special values at positive integers of those functions, including ζ W (2k; g), for some g explicitly (see [2,4,5,8]). …”

“…As mentioned above, the following definition is due to Matsumoto and Tsumura [8] and Komori et al [3]. Matsumoto and Tsumura [8] first introduced the sl(l) case.…”

On Witten Multiple Zeta-Functions Associated With Semi-Simple Lie Algebras Iv

“…For a real number x, let {x} denote its fractional part x − [x]. Applying Theorem 4.1 in [8] to the case of G 2 -type, we obtain Then F(t, y; G 2 ) is holomorphic at the origin and can be expanded as…”

“…ζ 2 (4, 4, 4, 4, 4, 4; G 2 ) = 8165653 1445838676129559305994400000 π 24 ; ζ 2 (6, 6, 6, 6, 6, 6; G 2 ) = 55940539974690617 131888156302530666544150214880458495963616000000 π 36 ; ζ 2 (8,8,8,8,8,8;…”

“…This type of generalised Bernoulli polynomials associated with any root system was first introduced in [10], and was further studied in [8]. For a real number x, let {x} denote its fractional part x − [x].…”