In the following $p$ is a prime number, $M(p)$ is a multiple of $p$, $\mu$ is an odd number. In the cited article (see below), page 2 (200), what properties are used to rewrite from line 4 to line 5? That is:
\begin{align} \frac{p}{2}\sum_{x=1}^{p-1}\frac{(p-x)^{\mu-1}-\ldots+x^{\mu-1}}{x^{\mu}(p-x)^{\mu}}=\frac{p}{2}\{M(p)-\frac{\mu}{2}\sum_{x=1}^{p-1}\frac{x^{\mu-1}}{x^{2\mu}}\} \end{align}
I might see a rewriting like the following for instance:
\begin{align} \ldots=\frac{p}{2}\{M(p)-\frac{\mu}{2}\sum_{x=1}^{p-1}\frac{x^{\mu-1}}{M(p)-x^{2\mu}}\} \end{align}
But how to move that $M(p)$ out of the denominator?
Leudesdorf, C., Some results in the elementary theory of numbers., Lond. M. S. Proc. XX, 199-212 (1889). ZBL21.0182.03.
Edit: notation unclear, not specified in reference, assume $x$ index for summation.