Some identities for multiple alternating zeta values

Abstract For any positive integers k , m , n with m ≥ 2 and k ≤ n , let A ( m , n , k ) be the sum of all the multiple alternating zeta values of weight mn and depth k with arguments being multiples of m, i.e., A ( m , n , k ) = ∑ s 1 + s 2 + … + s k = n s j ∈ N ζ ( m s 1 ‾ , m s 2 ‾ , … , m s k ‾ ) . In this note, we obtain the evaluation A ( m , n , k ) = ∑ a + b = n a , b ∈ N 0 ( − 1 ) a − k ( a k ) ⋅ ζ ( { m ‾ } a ) ⋅ ζ ⋆ ( { m ‾ } b ) , where ζ ( { m ‾ } a ) and ζ ⋆ ( { m ‾ } b ) denote the multiple alternating zeta values and multiple alternating zeta-star values respectively. Moreover, we give the evaluations of ζ ( { 2 m ‾ } n ) and ζ ⋆ ( { 2 m ‾ } n ) for arbitrary positive integers m , n .