# Polylog of root of unity

Calculus Level 5

$\large \sum_{k=1}^n \text{Li}_2(\omega^k x)=f(n)\text{Li}_2(x)+g(n)\text{Li}_2(x^n)$

The equation above holds true for rational functions $$f$$ and $$g$$ and $$\omega$$ is the primitive $$n^\text{th}$$ roots of unity for integer $$n>1$$. Find $$(f+g)(1)$$.

Notation: $${ \text{Li} }_{ n }(a)$$ denotes the polylogarithm function, $${ \text{Li} }_{ n }(a)=\displaystyle\sum _{ k=1 }^{ \infty }{ \frac { { a }^{ k } }{ { k }^{ n } } }.$$

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