We define Witten multiple zeta-functions associated with semisimple Lie algebras , of several complex variables, and prove the analytic continuation of them. These can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case , we determine the singularities of this function. Furthermore we prove certain functional relations among this function, the Mordell-Tornheim double zeta-functions and the Riemann zeta-function. Using these relations, we prove new and non-trivial evaluation formulas for special values of this function at positive integers.
Nous définissons les fonctions zeta multiples de Witten associées aux algèbres de Lie semi-simples , , et démontrons leurs continuations analytiques. Elles peuvent être considérées comme des généralisations à plusieurs variables des fonctions zeta de Witten définies par Zagier. Dans le cas , nous déterminons les singularités de la fonction zeta multiple. De plus, nous démontrons plusieurs relations fonctionnelles entre cette fonction, les fonctions zeta doubles de Mordell-Tornheim et la fonction zeta de Riemann. En utilisant ces relations, nous démontrons de nouvelles formules non-triviales pour évaluer des valeurs spécifiques de cette fonction aux points entiers positifs.
Keywords: Witten multiple zeta-functions, Mordell-Tornheim zeta-functions, Riemann zeta-function, analytic continuation, semisimple Lie algebra
Mot clés : fonctions zeta multiples de Witten, fonctions zeta de Mordell-Tornheim, fonctions zeta de Riemann, suite analytique, algèbre de Lie semisimple
@article{AIF_2006__56_5_1457_0, author = {Matsumoto, Kohji and Tsumura, Hirofumi}, title = {On {Witten} multiple zeta-functions associated with semisimple {Lie} algebras {I}}, journal = {Annales de l'Institut Fourier}, pages = {1457--1504}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {56}, number = {5}, year = {2006}, doi = {10.5802/aif.2218}, zbl = {1168.11036}, mrnumber = {2273862}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2218/} }
TY - JOUR AU - Matsumoto, Kohji AU - Tsumura, Hirofumi TI - On Witten multiple zeta-functions associated with semisimple Lie algebras I JO - Annales de l'Institut Fourier PY - 2006 SP - 1457 EP - 1504 VL - 56 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2218/ DO - 10.5802/aif.2218 LA - en ID - AIF_2006__56_5_1457_0 ER -
%0 Journal Article %A Matsumoto, Kohji %A Tsumura, Hirofumi %T On Witten multiple zeta-functions associated with semisimple Lie algebras I %J Annales de l'Institut Fourier %D 2006 %P 1457-1504 %V 56 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2218/ %R 10.5802/aif.2218 %G en %F AIF_2006__56_5_1457_0
Matsumoto, Kohji; Tsumura, Hirofumi. On Witten multiple zeta-functions associated with semisimple Lie algebras I. Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 1457-1504. doi : 10.5802/aif.2218. http://archive.numdam.org/articles/10.5802/aif.2218/
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