Polynomial overreproduction by Hermite subdivision operators, and $p$-Cauchy numbers

We study the case of Hermite subdivision operators satisfying a spectral condition of order greater than their size. We show that this can be characterized by operator factorizations involving Taylor operators and difference factorizations of a rank one vector scheme. Giving explicit expressions for the factorization operators, we put into evidence that the factorization only depends on the order of the spectral condition but not on the polynomials that define it. We further show that the derivation of these operators is based on an interplay between Stirling numbers and $p$-Cauchy numbers (or generalized Gregory coefficients).