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 depth respectively. Considerable amount of work on MZV’s has been done in recent years from various aspects and interests. Among them, several explicit values are known for special index sets, as will be recalled below. In this paper, we give some evaluations of themultiple zeta-star value (MZSV), which is defined by the following series similar to the MZV: ζ(k1, k2, . . . , kn) := ∑ m1≥m2≥···≥mn>0 1 m1 1 m k2 2 · · ·m kn n , where k1, k2, . . . , kn satisfy the same condition as above. The MZSV can be expressed as a Z-linear combination of MZV’s, and vice versa. Theorem A. For positive integers m,n, we have ζ(2m, 2m, · · · , 2m } {{ } n ) =    ∑ n0+···+nm−1=mn ni≥0 (−1) ( m−1 ∏ k=0 (2k − 2)B2nk (2nk)! ) exp ( 2πi m m−1 ∑ l=0 lnl )    π.