On Witten multiple zeta-functions associated with semisimple Lie algebras I
Annales de l'Institut Fourier, Volume 56 (2006) no. 5, p. 1457-1504

We define Witten multiple zeta-functions associated with semisimple Lie algebras 𝔰𝔩(n), (n=2,3,...) 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 𝔰𝔩(4), 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 𝔰𝔩(n), (n=2,3,...), 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 𝔰𝔩(4), 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.

DOI : https://doi.org/10.5802/aif.2218
Classification:  11M41,  40B05
Keywords: Witten multiple zeta-functions, Mordell-Tornheim zeta-functions, Riemann zeta-function, analytic continuation, semisimple Lie algebra
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {5},
     year = {2006},
     pages = {1457-1504},
     doi = {10.5802/aif.2218},
     mrnumber = {2273862},
     zbl = {1168.11036},
     language = {en},
     url = {http://www.numdam.org/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, Volume 56 (2006) no. 5, pp. 1457-1504. doi : 10.5802/aif.2218. http://www.numdam.org/item/AIF_2006__56_5_1457_0/

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