On Tornheim's double series

Abstract

arises in the study of the gamma function (see, for example, [3, p. 47]). The series T (r, s, t) is finite if and only if (1.1) r + t > 1, s+ t > 1, and r + s+ t > 2. The basic properties of T (r, s, t) (assuming (1.1)) are the following T (r, s, t) = T (s, r, t), (1.2) T (r, s, 0) = ζ(r)ζ(s), (1.3) T (r, 0, t) + T (t, 0, r) = ζ(r)ζ(t)− ζ(r + t), r ≥ 2, (1.4) T (r, s− 1, t+ 1) + T (r − 1, s, t+ 1) = T (r, s, t), r ≥ 1, s ≥ 1, (1.5) where ζ denotes the Riemann zeta function.