New Approximation method for the computation of particular values of polylogarithms

Abstract

An efficient procedure for the computation of $Li_{s}(z)$ where $s<0$ is here presented. We started with Polylogarithm $Li_{s}(z)$ where $s<0$. The summation of $n^{s}z^{n}$ is evaluated using a new method. An assumption is made that the power $n$ is multiplied by $x$ where $x=1$; then the series is integrated $s$ time in order to cancel the term $n^s$ out leaving the term $z^n$. By simply taking the derivative of the result $s$-times, an expression to evaluate the series arises which include only a constant term and the s$^{th}$ order of derivative of the summation of a simple geometric series. The computation of $Li_{s}(z)$ can then be performed in $\mathcal{O}(n)$ operations.