On a multiple harmonic power series.

Michel Emery (1)

1 IRMA - Institut de Recherche Mathématique Avancée

If $\Li_s$ denotes the polylogarithm of order~$s$, where $s$ is a natural number, and if $z$ belongs to the unit disk, \[ \Li_s {\Bigl({-z\over1\,{-}\,z}\Bigr)} =-\sum _{1\le i_1\le \ldots\le i_s} {z^{i_s}\over i_1i_2\ldots i_s}\;. \] In particular, \[ \sum _{n\ge 1}{\mup {n+1}\over n^s\!}\,= \sum _{1\le i_1\le \ldots\le i_s} {1\over{i_1}\ldots{i_s}\,2^{i_s}}\;. \]

Mots clés: Multiple harmonic series; Lerch function; Polylogarithm.; Polylogarithm

Domaines: Théorie des nombres [math.NT]

Michel Emery. On a multiple harmonic power series.. 2004. ⟨hal-00003259v1⟩