Here is an approach that leads to an integral equation for
$$A(\nu,t) := \sum_{n=0}^{\infty}(-1)^{n}\frac{a_{n}(\nu)}{\nu} e^{Int}.$$

First we notice that
$$(-1)^{n}\frac{a_{n}(\nu)}{\nu} = \frac{1}{2\pi} \int_0^{2\pi} A(\nu,t) e^{-Int}\,dt.$$

The given identity multiplied by $e^{2mz}$ can be restated as
\begin{split}
\frac{e^{2mz}}{m} &= \sum_{n=0}^{\infty}(-1)^{n}a_{n} (\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}e^{2mz} \\
&= (-1)^{m}\frac{a_m(\nu)e^{2mz}}{2m} + \sum_{n=0}^{m-1} (-1)^{n}\frac{a_{n} (\nu)}{\nu}\sin(\nu\,(m-n)) \frac{e^{(m-n)z}}{m-n}\frac{e^{(m+n)z}}{m+n} + \sum_{n=m+1}^{\infty} (-1)^{n}\frac{a_{n} (\nu)}{\nu}\sin(\nu\,(n-m)) \frac{e^{-(n-m)z}}{n-m}\frac{e^{(m+n)z}}{m+n}.
\end{split}

For the first sum, we have

\begin{split}
&\sum_{n=0}^{m-1} \frac{1}{2\pi}\int_0^{2\pi}dt\, A(\nu,t)e^{-Int} \Im e^{I\nu(m-n)} \int_{-\infty}^z dx\,e^{(m-n)x} \int_{-\infty}^z dy\,e^{(m+n)y} \\
=&\frac{1}{2\pi}\Im \int_0^{2\pi}dt\, A(\nu,t) \int_{-\infty}^z dx\, \int_{-\infty}^z dy\, e^{(I\nu+x+y)m} \sum_{n=0}^{m-1} e^{(-It-I\nu-x+y)n}\\
=&\frac{1}{2\pi}\Im \int_0^{2\pi}dt\, A(\nu,t) \int_{-\infty}^z dx\, \int_{-\infty}^z dy\, e^{(I\nu+x+y)m} \frac{1-e^{(-It-I\nu-x+y)m}}{1-e^{-It-I\nu-x+y}}\\
=&\frac{1}{2\pi}\Im \int_0^{2\pi}dt\, A(\nu,t) \int_{-\infty}^z dx\, \int_{-\infty}^z dy\, \frac{e^{(I\nu+x+y)m}-e^{(-It+2y)m}}{1-e^{-It-I\nu-x+y}}.
\end{split}

Similarly, for the second sum
\begin{split}
&\sum_{n=m+1}^{\infty} \frac{1}{2\pi}\int_0^{2\pi}dt\, A(\nu,t)e^{-Int} \Im e^{I\nu(n-m)} \int_{-\infty}^{-z} dx\,e^{(n-m)x} \int_{-\infty}^z dy\,e^{(m+n)y} \\
=&\frac{1}{2\pi}\Im \int_0^{2\pi}dt\, A(\nu,t) \int_{-\infty}^{-z} dx\, \int_{-\infty}^z dy\, e^{(-I\nu-x+y)m} \sum_{n=m+1}^{\infty} e^{(-It+I\nu+x+y)n}\\
=&\frac{1}{2\pi}\Im \int_0^{2\pi}dt\, A(\nu,t) \int_{-\infty}^{-z} dx\, \int_{-\infty}^z dy\, e^{(-I\nu-x+y)m} \frac{e^{(-It+I\nu+x+y)(m+1)}}{1-e^{-It+I\nu+x+y}}\\
=&\frac{1}{2\pi}\Im \int_0^{2\pi}dt\, A(\nu,t) \int_{-\infty}^{-z} dx\, \int_{-\infty}^z dy\, e^{-It+I\nu+x+y} \frac{e^{(-It+2y)m}}{1-e^{-It+I\nu+x+y}}.
\end{split}

Finally,
$$(-1)^{m}\frac{a_m(\nu)e^{2mz}}{2m} = \frac{\nu}{2\pi}\int_0^{2\pi}dt\, A(\nu,t) \int_{-\infty}^z dy\, e^{(-It+2y)m}.$$

Summing over $m=1,2,\dots$, we get
\begin{split}
-\log(1-e^{2z}) &= \frac{\nu}{2\pi}\int_0^{2\pi}dt\, A(\nu,t) \int_{-\infty}^z dy\, \frac{e^{-It+2y}}{1-e^{-It+2y}} \\
&+\frac{1}{2\pi}\Im \int_0^{2\pi}dt\, A(\nu,t) \int_{-\infty}^z dx\, \int_{-\infty}^z dy\, \frac{ \frac{e^{I\nu+x+y}}{1-e^{I\nu+x+y}}-\frac{e^{-It+2y}}{1-e^{-It+2y}}}{1-e^{-It-I\nu-x+y}}\\
&+\frac{1}{2\pi}\Im \int_0^{2\pi}dt\, A(\nu,t) \int_{-\infty}^{-z} dx\, \int_{-\infty}^z dy\, \frac{e^{-2It+I\nu+x+3y}}{(1-e^{-It+I\nu+x+y})(1-e^{-It+2y})}.
\end{split}

Evaluating integrals over $x$ and $y$ (which is a techincal task) will give an integral equation for $A(\nu,t)$.

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