Équations fonctionnelles du dilogarithme  [ Functional equations for Rogers dilogarithm ]
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, p. 151-169
This paper proves a “new” family of functional equations (Eq n ) for Rogers dilogarithm. These equations rely on the combinatorics of dihedral coordinates on moduli spaces of curves of genus 0, 0,n . For n=4 we find back the duality relation while n=5 gives back the 5 terms relation. It is then proved that the whole family reduces to the 5 terms relation. In the author’s knownledge, it is the first time that an infinite family of functional equations for the dilogarithm with an increasing number of variables (n-3 for (Eq n )) is reduced to the 5 terms relation.This reduction explains the quotation marks around “new” at the beginning of this abstract.
Cet article démontre une « nouvelle » famille d’équations fonctionnelles (Eq n ) (n4) satisfaites par le dilogarithme de Rogers. Ces équations fonctionnelles reflètent la combinatoire des coordonnées diédrales des espaces de modules de courbes de genres 0, 0,n . Pour n=4, on retrouve la relation de dualité et, pour n=5, la relation à 5 termes du dilogarithme. Dans une seconde partie, on démontre que la famille (Eq n ) se réduit à la relation à 5 termes. C’est, à la connaissance de l’auteur, la première fois qu’une famille infinie d’équations fonctionnelles du dilogarithme ayant un nombre croissant de variables (n-3 pour (Eq n )) se réduit à la relation à 5 termes.La réduction de cette famille d’équations à la relation de 5-cycle explique les guillemets de la première phrase.
Received : 2015-11-02
Revised : 2017-05-28
Accepted : 2017-09-14
Published online : 2018-04-18
DOI : https://doi.org/10.5802/aif.3155
Classification:  39B50,  11G55,  39B22,  39B32,  33B30
Keywords: functional equations, dilogarithm, polylogarithms, moduli spaces, modulispaces of curves of genus 0
@article{AIF_2018__68_1_151_0,
     author = {Soud\`eres, Ismael},
     title = {\'Equations fonctionnelles du dilogarithme},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     pages = {151-169},
     doi = {10.5802/aif.3155},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_1_151_0}
}
Équations fonctionnelles du dilogarithme. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 151-169. doi : 10.5802/aif.3155. https://aif.centre-mersenne.org/item/AIF_2018__68_1_151_0/

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