CARMA ANALYSIS AND NUMBER THEORY SEMINAR Speaker: Laureate Prof Jon Borwein, CARMA, The University of Newcastle Title: Computation and theory of extended Mordell-Tornheim-Witten sums I Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 1:00 pm, Tue, 1st May 2012 Abstract: We consider some fundamental generalized Mordell-Tornheim-Witten (MTW) zeta-function values along with their derivatives, and explore connections with multiple-zeta values (MZVs). To achieve these results, we make use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function theory. Our original motivation was to represent previously unresolved constructs such as Eulerian log-gamma integrals. Indeed, we are able to show that all such integrals belong to a vector space over an MTW basis, and we also present, for a substantial subset of this class, explicit closed-form expressions. In the process, we significantly extend methods for high-precision numerical computation of polylogarithms and their derivatives with respect to order. That said, the focus of our paper is the relation between MTW sums and classical polylogarithms. It is the adumbration of these relationships that makes the study significant. The associated paper (with DH Bailey and RE Crandall) is at http://carmasite.newcastle.edu.au/jon/MTW1.pdf. [Permanent link]