Abstract <p>The Vandermonde matrix is considered in an arbitrary complex Banach algebra. The accompanying Frobenius matrix is used to establish the relationship between the coefficients of an algebraic equation and the Vandermonde matrix constructed from its roots. The divided difference of arbitrary order is defined based on an invertible Vandermonde matrix. An analogue of the Hermite formula for an integral representation of the divided difference is given. An inclusion for the spectrum of the divided difference and an analogue of Dunford’s theorem on the mapping of spectra are given.</p>