Various single-valued versions of ordinary polylogarithms Lin(z) have been constructed by Ramakrishnan, Wojtkowiak, Zagier, and others. These single-valued functions are generalisations of the Bloch–Wigner dilogarithm and have many applications in mathematics. In this Note we show how to construct explicit single-valued versions of multiple polylogarithms in one variable. We prove the functions thus constructed are linearly independent, that they satisfy the shuffle relations, and that every possible single-valued version of multiple polylogarithms in one variable can be obtained in this way.
Les versions uniformes des polylogarithmes classiques Lin(z) ont de nombreuses applications en mathématiques. Elles ont été étudiées, sous diverses formes, par Ramakrishnan, Wojtkowiak, et Zagier entre autres. Dans cette Note nous expliquons comment construire, plus généralement, une version uniforme pour chaque polylogarithme multiple en une variable de façon systématique. Nous démontrons que les fonctions définies sont linéairement indépendantes, qu'elles satisfont aux relations de battage, et que toute version uniforme des polylogarithmes multiples en une variable s'obtient de cette façon.
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@article{CRMATH_2004__338_7_527_0, author = {Brown, Francis C.S.}, title = {Polylogarithmes multiples uniformes en une variable}, journal = {Comptes Rendus. Math\'ematique}, pages = {527--532}, publisher = {Elsevier}, volume = {338}, number = {7}, year = {2004}, doi = {10.1016/j.crma.2004.02.001}, language = {fr}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2004.02.001/} }
TY - JOUR AU - Brown, Francis C.S. TI - Polylogarithmes multiples uniformes en une variable JO - Comptes Rendus. Mathématique PY - 2004 SP - 527 EP - 532 VL - 338 IS - 7 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2004.02.001/ DO - 10.1016/j.crma.2004.02.001 LA - fr ID - CRMATH_2004__338_7_527_0 ER -
%0 Journal Article %A Brown, Francis C.S. %T Polylogarithmes multiples uniformes en une variable %J Comptes Rendus. Mathématique %D 2004 %P 527-532 %V 338 %N 7 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2004.02.001/ %R 10.1016/j.crma.2004.02.001 %G fr %F CRMATH_2004__338_7_527_0
Brown, Francis C.S. Polylogarithmes multiples uniformes en une variable. Comptes Rendus. Mathématique, Volume 338 (2004) no. 7, pp. 527-532. doi : 10.1016/j.crma.2004.02.001. http://archive.numdam.org/articles/10.1016/j.crma.2004.02.001/
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