On Witten multiple zeta-functions associated with semisimple Lie algebras I

Matsumoto, Kohji; Tsumura, Hirofumi

Annales de l'Institut Fourier, Tome 56 (2006) no. 5, pp. 1457-1504.

Nous définissons les fonctions zeta multiples de Witten associées aux algèbres de Lie semi-simples $\mathrm{𝔰𝔩}\left(n\right)$, $(n=2,3,...)$, 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 $\mathrm{𝔰𝔩}\left(4\right)$, 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 $\mathrm{𝔰𝔩}\left(n\right)$, $(n=2,3,...)$ 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 $\mathrm{𝔰𝔩}\left(4\right)$, 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.

Classification : 11M41, 40B05
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

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