Recurrence Relation for Polylogarithms

Theorem

$\ds \map {\Li_{s + 1} } z = \int_0^z \dfrac {\map {\Li_s} t} t \rd t$

where:

$\map {\Li_s} z$ denotes the polylogarithm.

$\ds \int_0^z \dfrac {\map {\Li_s} t} t \rd t$

$=$

$\ds \int_0^z \frac 1 t \times \sum_{n \mathop = 1}^\infty \frac {t^n} {n^s} \rd t$

Definition of Polylogarithm

$\ds $

$=$

$\ds \int_0^z \sum_{n \mathop = 1}^\infty \frac {t^{n - 1} } {n^s} \rd t$

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty \int_0^z \frac {t^{n - 1} } {n^s} \rd t$

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty \intlimits {\frac {t^n} {n^{s + 1} } } 0 z$

$\ds $

$=$

$\ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n^{s + 1} }$

$\ds $

$=$

$\ds \map {\Li_{s + 1} } z$

Definition of Polylogarithm

$\blacksquare$