  We prove some functional equations involving the (classical) matching
polynomials of path and cycle graphs and the $d$-matching polynomial of a cycle
graph. A matching in a (finite) graph $G$ is a subset of edges no two of which
share a vertex, and the matching polynomial of $G$ is a generating function
encoding the numbers of matchings in $G$ of each size. The $d$-matching
polynomial is a weighted average of matching polynomials of degree-$d$ covers,
and was introduced in a paper of Hall, Puder, and Sawin. Let $\mathcal{C}_n$
and $\mathcal{P}_n$ denote the respective matching polynomials of the cycle and
path graphs on $n$ vertices, and let $\mathcal{C}_{n,d}$ denote the
$d$-matching polynomial of the cycle $C_n$. We give a purely combinatorial
proof that $\mathcal{C}_k (\mathcal{C}_n (x)) = \mathcal{C}_{kn} (x)$ en route
to proving a conjecture made by Hall: that $\mathcal{C}_{n,d} (x) =
\mathcal{P}_d (\mathcal{C}_n (x))$.
