Computation and theory of Mordell–Tornheim–Witten sums II

Abstract

In [8] the current authors, along with the late and much-missed Richard Crandall , considered generalized Mordell-Tornheim-Witten (MTW) zeta-function values along with their derivatives, and explored connections with multiple-zeta values (MZVs). This entailed use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function theory. The original motivation was to represent objects such as Eulerian log-gamma integrals; and all such integrals were expressed in terms of a MTW basis. Herein, we extend the research envisaged in [8] by analyzing the relations between a significantly more general class of MTW sums. This has required significantly more subtle scientific computation and concomitant special function theory.

The Polylogarithm Function in Julia

“…• Crandall [8] presents a unified version of his earlier approach for general s (and other related functions), as well as the expression for the expansions near negative integer s, but in this work does not report a complete algorithm, and also suffers from some small mistakes. • Bailey and Borwein [3,4] use and refine Crandall's work for calculating polylogarithms and their derivatives. They discuss alternative approaches in different domains, but their main interest is in related functions and although they fill in some gaps of [8], they do not present a complete algorithm either.…”

“…and gives the c k,j in [8, pp.35-36] recursively, but note that Crandall's manuscript has typographic errors; Bailey and Borwein [3,4] give the correct formula. However, for |τ | 1 we need only take a small number of terms, the first three of which can be written explicitly as 2) (n + 1) 6 .…”

“…There are many publications on the function, but only a handful on its numerical evaluation [10,23,7,19,8,3,4]. Zagier [24] precisely describes the situation "It occurs not quite often enough, and in not quite an important enough way, to be included in the Valhalla of the great transcendental functions-the gamma function, Bessel and Legendre-functions, hypergeometric series, or Riemann's zeta function.…”