Für alle $n\in\N$ gilt:
$$
n^2\Phi^\dprime(nz)=\sum_{k=0}^{n-1}\Phi^\dprime(z+\tfrac kn)~.
$$.
\begin{eqnarray*}
n^2\Phi^\dprime(nz)
&=&\sum_{j=0}^\infty\frac{n^2}{(nz+j)^2}
=\sum_{j=0}^\infty\frac{1}{(z+j/n)^2}\\
&=&\sum_{j=0}^\infty\sum_{k=0}^{n-1}\frac{1}{(z+\frac{jn+k}n)^2}
=\sum_{k=0}^{n-1}\sum_{j=0}^\infty\frac{1}{(z+\frac kn+j)^2}
=\sum_{k=0}^{n-1}\Phi^\dprime(z+\tfrac kn)
\end{eqnarray*}