Are there some known identities of elliptic polylogarithms similar to the Abel identity of polylogarithm? 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.

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).
