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Archiv der Mathematik

, Volume 88, Issue 1, pp 42–51 | Cite as

On certain polylogarithmic double series

  • Hirofumi Tsumura
Article

Abstract.

In this paper, we study certain polylogarithmic double series
$$ {\sum\limits_{m, n = 1}^\infty {\frac{{x^{n} }} {{m^{p} n^{q} (m + n)^{r} }}} },{\sum\limits_{m, n = 1}^\infty {\frac{{x^{{m + n}} }} {{m^{p} n^{q} (m + n)^{r} }}} }, $$
where p, q, r are nonnegative integers and x is any complex number with
$$ {\left| x \right|} \leq 1 $$
. First we give certain polylogarithmic interpolation formulas of the results of Mordell, Subbarao-Sitaramachandrarao and Zagier. By specializing to x = 1, we can obtain their results. Secondly we calculate some special values of these polylogarithmic double series.

Mathematics Subject Classification (2000).

Primary 40B05 Secondary 11M06, 30B99 

Keywords.

Double polylogarithms Tornheim’s double sums Riemann’s zeta-function 

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Copyright information

© Birkhäuser Verlag, Basel/Switzerland 2006

Authors and Affiliations

  1. 1.Department of Mathematics and Information SciencesTokyo Metropolitan UniversityTokyoJapan

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