Consider a graph $G$ with $n$ vertices. Let $m\_k$ be the number of $k$-edge [matchings](http://en.wikipedia.org/wiki/Matching_%28graph_theory%29). (A matching in a graph is a set of edges without common vertices.) Several different types of [matching polynomials](http://en.wikipedia.org/wiki/Matching_polynomial) have been defined, but consider the following definition.

\begin{equation\*}
M\_G(x) = \sum\_{0\leq k}(-1)^k m\_k x^{n-2k}
\end{equation\*}

According to the wikipedia article, the matching polynomial for $G=K\_{m,n}$ ([a complete bipartite graph](http://en.wikipedia.org/wiki/Complete_bipartite_graph)) is related to the [generalized Laguerre polynomial](http://en.wikipedia.org/wiki/Laguerre_polynomials) $L\_n^{\alpha}(x)$:

\begin{equation\*}
M\_{K\_{m,n}} = n! L\_n^{(m-n)}(x^2)
\end{equation\*}

How can I prove it? Is there any reference (paper or textbook) so that I can derive this relation myself?