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

## TL;DR

This paper characterizes Hermite subdivision operators satisfying high-order spectral conditions through operator factorizations involving Taylor and difference operators, revealing a connection with Stirling and $p$-Cauchy numbers.

## Contribution

It provides explicit factorizations of Hermite subdivision operators based on spectral order, independent of defining polynomials, linking them to Stirling and $p$-Cauchy numbers.

## Key findings

- Operator factorizations depend only on spectral order.
- Explicit expressions for factorization operators are derived.
- The derivation involves Stirling and $p$-Cauchy numbers.

## Abstract

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).