EXPLICIT EVALUATION OF TRIPLE EULER SUMS

Abstract

The triple Euler sum defined by [Formula: see text] with positive integers p, q, r, p ≥ 2, has not been fully evaluated yet except for the case of q = r = 1 and some particular cases such as p + q + r ≤ 10 or p = q = r. With the general theory developed for double Euler sums, we are able to produce identities among triple Euler sums with a variable or two when q = r =. Performing differentiations with respect to the specified variable gives the explicit evaluations of ζ(p, q, r) for general p, q, r. In particular, the values of ζ(n, 1, 1), ζ(2n + 1, 1, 2), ζ(2n + 1, 2, 1) and ζ(2n, 2, 2) are given explicitly in terms of classical double Euler sums. Also ζ(2n, 1, 2m + 1) and ζ(2n + 1, 1, 2m) are obtained for all positive integers m and n.