Some alternating double sum formulae of multiple zeta values

Abstract

In this paper, we produce shuffle relations from multiple zeta values of the form ζ ({ 1 }m-1, n+1). Here { 1 }k is k repetitions of 1, and for a string of positive integers α1, α2, ...,αr with αr > 2 . ζ (α1, α1, ..., α1) = Σ n1-α1n2-α2... n r-αr 1 < n1 < n2 < ... < n r As applications of the sum formula and a newly developed weighted sum formula, we shall prove for even integers k, r > 0 that k r Σ Σ (-1)ℓ Σ ζ (α0, α1, ..., αj + βj, βj+1, ..., βk, βk+1 + 1) j = 0 ℓ = 0 |α| = j + r - ℓ + 1 |β| = k - j + ℓ + 2 + Σ Σ ζ (α0, α1, ..., αk r - ℓ + 3) = ζ (k + r + 4). 0 < ℓ < r |α| = k + ℓ + 1 ℓ : even Mathematical subject classification: Primary: 40A25, 40B05; Secondary: 11M99, 33E99.