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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