Evaluation of an analytical function of a companion matrix with multiple eigenvalues

An analytical function of a companion matrix with multiple eigenvalues is expressed as the linear combination of constituent matrices. It is found that the constituent matrix is simply the product of the two matrices whine elements are part of the entries of Vandermonde matrix and its inverse. Calculation of these values is given as simple, explicit, and recunent formulas, and is very useful for both hand and machine computations.