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. 1

Summary. This introductory article aims to provide a roadmap to many of the

Abstract. In this paper we shall prove that every Witten multiple zeta value of weight w> 3 attached to sl(4) at nonnegative integer arguments is a finite Q-linear combination of MZVs of weight w and depth three or less, except for the nine irregular cases where the Riemann zeta value ζ(w − 2) and the double zeta values of weight w − 1 and depth < 3 are also needed. 1.

Abstract. In this note we shall study the Witten multiple zeta function associated to the Lie algebra so(5) defined by Matsumoto. Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight w ≥ 3 is a finite rational linear combination of alternating Euler sums of weight w and depth at most two, except when the only nonzero argument is one of the two last variables in which case ζ(w − 1) is needed. 1.

Abstract. The shuffle product plays an important role in the study of multiple zeta values. This is expressed in terms of multiple integrals, and also as a product in a certain non-commutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of the shuffle product. In fact, we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of multiple zeta values of root systems. As an application, we give a proof of extended double shuffle relations without using Drinfel’d integral expressions for multiple zeta values. Furthermore, our argument enables us to give some functional relations which include double shuffle relations.