The classical Euler sum [Formula: see text] cannot be evaluated when the weight p + q is even unless p = 1 or p = q or (p, q) = (2, 4) or (p, q) = (4, 2) [7]. However it is a different story if instead we consider the alternating sums [Formula: see text] and [Formula: see text] They can be evaluated for even weight p + q. In this paper, we shall evaluate a family of generalized Euler sums containing [Formula: see text] when the weight p + q is even via integral transforms of Bernoulli identities.

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 α 1 , α 2 , . . . , α r with α r ≥ 2As applications of the sum formula and a newly developed weighted sum formula, we shall prove for even integers k, r ≥ 0 thatMathematical subject classification: Primary: 40A25, 40B05; Secondary: 11M99, 33E99.

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