New Approximation method for the computation of particular values of polylogarithms

TL;DR

This paper introduces a new efficient method for computing polylogarithms with negative s by transforming the series through integration and differentiation, reducing computational complexity to linear time.

Contribution

The paper presents a novel approach that simplifies the computation of Li_s(z) for s<0 using integration and differentiation, improving efficiency.

Findings

Method reduces computation to O(n) operations.

Series transformation simplifies evaluation of polylogarithms.

Approach is applicable for negative s values.

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.