Polylogarithmes multiples uniformes en une variable

Abstract

Various single-valued versions of ordinary polylogarithms Li n ( 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. To cite this article: F.C.S. Brown, C. R. Acad. Sci. Paris, Ser. I 338 (2004).