# Matching polynomial of a complete bipartite graph is a generalized Laguerre polynomial.

Consider a graph $G$ with $n$ vertices. Let $m_k$ be the number of $k$-edge matchings. (A matching in a graph is a set of edges without common vertices.) Several different types of matching polynomials 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) is related to the generalized Laguerre polynomial $L_n^{\alpha}(x)$:

\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?