Polylogarithm recursion identity

I recently discovered a recursive identity for \(\mathrm{Li}_{-m}(z)\) for \(m\) a positive integer. Hopefully this will be of some use when tackling problems seeking for the value of some polylogarithmic sum.

\[\mathrm{Li}_{-m}(z) = \frac{z}{1-z} \left[ 1+\sum_{k=0}^{m-1} \binom{m}{k} \mathrm{Li}_{-k}(z) \right]\]


Proof:

Consider \(\displaystyle \sum_{n=1}^{\infty} z^n(n+1)^m\). We have

\[\begin{align} \sum_{n=1}^{\infty} z^n(n+1)^m &= \sum_{n=1}^{\infty} z^n \sum_{k=0}^m \binom{m}{k} n^k \\ &= \sum_{n=1}^{\infty} z^nn^m + \sum_{k=0}^{m-1} \binom{m}{k} \sum_{n=1}^{\infty} z^nn^k \\ &= \mathrm{Li}_{-m}(z) + \sum_{k=0}^{m-1} \binom{m}{k} \mathrm{Li}_{-k}(z) \end{align}\]

We also have

\[\begin{align} \sum_{n=1}^{\infty} z^n(n+1)^m &= \frac{1}{z} \sum_{n=2}^{\infty} z^nn^m \\ &= \frac{1}{z} \mathrm{Li}_{-m}(z)-1 \end{align}\]

Equating the two gives

\[\mathrm{Li}_{-m}(z) + \sum_{k=0}^{m-1} \binom{m}{k} \mathrm{Li}_{-k}(z) = \frac{1}{z} \mathrm{Li}_{-m}(z)-1\]

When rearranged, we obtain

\[\mathrm{Li}_{-m}(z) = \frac{z}{1-z} \left[ 1+\sum_{k=0}^{m-1} \binom{m}{k} \mathrm{Li}_{-k}(z) \right]\]

as required.

Note by Jake Lai
2 years, 8 months ago

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Similarly \[\mathrm{Li}_{-m}(z)=\frac{\left(-1\right)^m}{1-z}\left[\sum _{k=0}^{m-1}\left(-1\right)^k\binom{m}{k}\mathrm{Li}_{-k}(z)\right]\]

Julian Poon - 2 years, 8 months ago

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Nice!

Jake Lai - 2 years, 8 months ago

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A point to note is that \(\operatorname{Li}_{-m} (z)\) , when \(m\) is a positive integer, already has a closed form in terms of rational polynomial functions.

Ishan Singh - 2 years, 2 months ago

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