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.
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.
@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}, volume = {56}, year = {2006}, pages = {1457-1504}, doi = {10.5802/aif.2218}, zbl = {1168.11036}, mrnumber = {2273862}, language = {en}, url = {http://dml.mathdoc.fr/item/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, Tome 56 (2006) pp. 1457-1504. doi : 10.5802/aif.2218. http://gdmltest.u-ga.fr/item/AIF_2006__56_5_1457_0/
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