Bulletin of the Australian Mathematical Society

Table of Contents - December 2012 - Volume 86, Issue 03  

Research Article

ON A CERTAIN CONVOLUTION OF POLYLOGARITHMS

HIROFUMI TSUMURAa1

a1 Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan (email: tsumura@tmu.ac.jp)

Abstract

In this paper, we consider certain double series analogous to Tornheim’s double series and real analytic Eisenstein series. By computing double integrals in two ways, we express the double series as a sum of products of polylogarithms. The technique generalises one given by Kanemitsu, Tanigawa and Yoshimoto. Evaluating the double series at particular points gives new evaluations for certain double series in terms of values of the Riemann zeta function and the dilogarithm which are analogues of formulas of Mordell and Goncharov.

(Received November 22 2011)

2010 Mathematics subject classification

  • primary 11M41; secondary 11M99

Keywords and phrases

  • polylogarithms;
  • Riemann zeta-function;
  • real analytic Eisenstein series

Footnotes

This research was partially supported by Grant-in-Aid for Science Research (No. 23540022), Japan Society for the Promotion of Science.