Polylogarithm of Square
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Theorem
- $\map {\Li_s} z + \map {\Li_s} {-z} = 2^{1 - s} \map {\Li_s} {z^2}$
where $\Li_s$ denotes the polylogarithm.
Proof
| \(\ds \map {\Li_s} z + \map {\Li_s} {-z}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n^s} + \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} {n^s}\) | Definition of Polylogarithm | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {z + \frac {z^2} {2^s} + \frac {z^3} {3^s} + \frac {z^4} {4^s} + \frac {z^5} {5^s} + \frac {z^6} {6^s} + \cdots} + \paren {-z + \frac {z^2} {2^s} - \frac {z^3} {3^s} + \frac {z^4} {4^s} - \frac {z^5} {5^s} + \frac {z^6} {6^s} + \cdots}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \paren {\frac {z^2} {2^s} + \frac {z^4} {4^s} + \frac {z^6} {6^s} + \cdots}\) | odd terms cancel, even terms double | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 2 {2^s} \paren {\frac {z^2} {1^s} + \frac {z^4} {2^s} + \frac {z^6} {3^s} + \cdots}\) | factoring out $2^s$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{1 - s} \sum_{n \mathop = 1}^\infty \frac {\paren {z^2}^n} {n^s}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{1 - s} \map {\Li_s} {z^2}\) | Definition of Polylogarithm |
$\blacksquare$