  The classical Hurwitz numbers count the fixed-length transitive transposition
factorizations of a permutation, with a remarkable product formula for the case
of minimum length (genus $0$). We study the analogue of these numbers for
reflection groups with the following generalization of transitivity: say that a
reflection factorization of an element in a reflection group $W$ is full if the
factors generate the whole group $W$. We compute the generating function for
full factorizations of arbitrary length for an arbitrary element in a group in
the combinatorial family $G(m, p, n)$ of complex reflection groups in terms of
the generating functions of the symmetric group $\mathfrak{S}_n$ and the cyclic
group of order $m/p$. As a corollary, we obtain leading-term formulas which
count minimum-length full reflection factorizations of an arbitrary element in
$G(m,p,n)$ in terms of the Hurwitz numbers of genus $0$ and $1$ and
number-theoretic functions. We also study the structural properties of such
generating functions for any complex reflection group; in particular, we show
via representation-theoretic methods that they can by expressed as finite sums
of exponentials of the variable.
