  In this paper we present a factorization framework for Hermite subdivision
schemes refining function values and first derivatives, which satisfy a
spectral condition of high order. In particular we show that spectral order $d$
allows for $d$ factorizations of the subdivision operator with respect to the
Gregory operators: A new sequence of operators we define using Stirling numbers
and Gregory coefficients. We further prove that the $d$-th factorization
provides a ``convergence from contractivity'' method for showing
$C^d$-convergence of the associated Hermite subdivision scheme. The power of
our factorization framework lies in the reduction of computational effort for
large $d$: In order to prove $C^d$-convergence, up to now, $d$ factorization
steps were needed, while our method requires only one step, independently of
$d$. Furthermore, in this paper, we show by an example that the spectral
condition is not equivalent to the reproduction of polynomials.
