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

On value-relations, functional relations and singularities of Mordell–Tornheim and related triple zeta-functions

“…From the assumption of induction, we have Using (6.20), we see that (6.23) and (6.26) determine true singularities, because (6.21) and (6.24) do. This argument for (6.23) and (6.26) is much simpler than the original method in [9]. Hence we can see that this kind of argument using symmetry is convenient for checking whether singularities are true or not.…”

Zeta Functions of Root Systems

“…For example, we can give a functional relation including (see [3]) as a special value-relation. As mentioned above, we were unable to treat Witten's volume formulas in our previous paper ( [12]), because we were only able to consider some limited types of relations for ζ 3 , that is, the case one-variable is equal to 0. In order to remove this limitation, we will study certain multiple polylogarithms of type A 3 (see (4.18) and (4.19)).…”

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

“…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. Afterwards, Komori et al [3] introduced the other cases (also see [2]).…”

“…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]). …”