# Supernumary Polylogarithmic Ladders and Related Functional Equations

• L. Lewin
Conference paper
As discussed in several recent papers(1–7) the polylogarithmic function of order n and argument z can be defined through the series
$$L{i_n}\left( z \right) = \sum\limits_{r = 1}^\infty {{z^r}} /{r^n},\left| z \right|1$$
(1.1)
It satisfies the recursion formula
$$L{i_n}\left( z \right) = \int\limits_0^z {L{i_{n - 1}}} \left( {z'} \right)dz'/z'$$
(1.2)
and this, together with the elementary relation
$$L{i_n}\left( z \right) = - \log \left( {1 - z} \right)$$
(1.3)
extends the definition throughout the complex z-plane.

## Keywords

Functional Equation Real Root Algebraic Number Inversion Formula Numerical Search
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