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Let \begin{align} Li_2(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^2}. \end{align} This polylogarithm satisfies the following Abel identity: \begin{align} & Li_2(-x) + \log x \log y \\ & + Li_2(-y) + \log ( \frac{1+y}{x} ) \log y \\ & + Li_2(-\frac{1+y}{x}) + \log ( \frac{1+y}{x} ) \log (\frac{1+x+y}{xy}) \\ & + Li_2(-\frac{1+x+y}{xy}) + \log ( \frac{1+x}{y} ) \log (\frac{1+x+y}{xy}) \\ & + Li_2(-\frac{1+x}{y}) + \log ( \frac{1+x}{y} ) \log x \\ & = - \frac{\pi^2}{2}. \end{align} The following function \begin{align} ELi_{n,m}(x,y,q) = \sum_{j=1}^{\infty} \sum_{k=1}^{\infty} \frac{x^j}{j^n} \frac{y^k}{k^m} q^{jk} \end{align} is defined in the paper in (2.1).

Are there some known identities similar to the Abel identity for $ELi_{n,m}(x,y,q)$? Thank you very much.

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  • $\begingroup$ Are there some conditions for x, and y ? for instance this paper disscussed many polylogharithms may it help :arxiv.org/pdf/hep-th/9408113.pdf , because i think your class of polylogarithms is of : Dilogarithm identities and conformal weights $\endgroup$ Feb 9, 2017 at 17:43

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The 5-term relation is a special case of the Rogers identity: theorem 8.14 here. This is a degenerate version of the Bloch relation for elliptic dilogarithm (see page 30 here).

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