The Multiple Summation Formula and Polylogarithms

Milorad Stevanović

Mathematica Moravica, Tome 9 (2005) no. 1.

Résumé

In this paper is given the formula: \[F_{n}(x) = um_{k_{1}=1}^{ıfty}\frac{x^{k_{1}}}{k_{1}}um_{k_{2}=1}^{k_{1}}\frac{x^{k_{2}}}{k_{2}}\cdots um_{k_{n}=1}^{k_{n-1}}\frac{x^{k_{n}}}{k_{n}} = um_{um_{j=1}^{n}j\cdotlpha_{j}=n, lpha_{j}\geq 0} \frac{rod_{k=1}^{n}\zeta_{k}^{lpha_{k}}(x^{k})}{rod_{k=1}^{n}k_{lpha_{k}}lpha_{k}!}\] \[n\geq 1,\quad -1eq x 1\] with \[\zeta_{k}(x) \equiv Li_{k}(x)\equiv um_{r=1}^{ıfty}\frac{x^{r}}{r^{k}},\qquad (k\geq 0),\] and method by which it can be obtained.

Détail

Mots-clés : multiple sums, multiple summation, polylogarithms