Hermite interpolation sequences over fields

By a Hermite interpolation sequence we mean a sequence of Hermite interpolation polynomials of degree 0,1,… such that consecutive terms satisfy the differentiation conditions of the previous ones. We extend this concept to arbitrary fields from the reals by purely algebraic means based on the possibility of formal Taylor expansions of rational fractions around any point of the underlying field. As an application we obtain recursion-free explicit formulas for the entries of triangular decompositions of generalized Hermite–Vandermonde matrices.