Multiple zeta star values on 3–2–1 indices

On some explicit evaluations of multiple zeta-star values

In this paper, we give some explicit evaluations of multiple zeta-star values which are rational multiple of powers of π. 1 Main Results The multiple zeta value (MZV) is defined by the convergent series ζ(k1, k2, . . . , kn) := ∑ m1>m2>···>mn>0 1 m1 1 m k2 2 · · ·m kn n , where k1, k2, . . . , kn are positive integers and k1 ≥ 2. The integers k = k1+k2+ · · ·+kn and n are called weight and dept...

Aspectsof Multiple Zeta Values

Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuue product rule allows the possibility of a combi-natorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with ...

Multiple Zeta Values

for any collection of positive integers s1, s2, . . . , sl. By definition, Lis(1) = ζ(s), s ∈ Z, s1 ≥ 2, s2 ≥ 1, . . . , sl ≥ 1. (4.2) Taking, as before for multiple zeta values, Lixs(z) := Lis(z), Li1(z) := 1, (4.3) let us extend action of the map Li : w 7→ Liw(z) by linearity on the graded algebra H (not H, since multi-indices are coded by words in H). Lemma 4.1. Let w ∈ H be an arbitrary non...

Multiple Zeta Values 17

It is now a good time to go back to the MZV story. where F (a, b; c; z) denotes the hypergeometric function and i = √ −1. Proof. Routine verification (with a help of Lemma 4.1 for the left-hand side) shows that the both sides of the required equality are annihilated by action of the differential operator (1 − z) d dz 2 z d dz 2 − t 4 ; in addition, the first terms in z-expansions of the both si...

Multiple Zeta Values