On hermite interpolation by Cauchy-Vandermonde systems: The Lagrange formula, the adjoint and the inverse of a Cauchy-Vandermonde matrix

For a given Cauchy-Vandermonde system and for given multiple nodes a Lagrange-type formula for the interpolant is derived, interpolating a given function in the sense of Hermite. We give explicit analytic representations of the basic functions in terms of the nodes and prescribed poles. They are used to derive formulas for the entries of the adjoint of the confluent Cauchy-Vandermonde matrix corresponding to the interpolation problem thus providing an explicit representation of its inverse.