Divided difference table from a matrix viewpoint

Given a complex function w=f(z) defined in some region containing distinct points z₁, ..., z_n, we consider the divided difference table in the form of the divided difference matrix f⁺(z₁, ..., z_n) whose (i, j) component equals f(z_i, ..., z_j) for i≤j and 0 elsewhere. Two theorems are proved: the first asserts that f→f⁺ is an algebraic homomorphism; the second gives a Cauchy contour-integral representation of f⁺(z₁, ..., z_n), which also equals the Cauchy formula for f(z⁺), where z⁺ denote the f⁺-matrix corresponding to f(z)=z and where f(z) is assumed analytic in a region containing z₁, ..., z_n.