A constructive formula for function of matrix. Alternative to the Lagrange–Sylvester formula

Instead of the elegant function (Lagrange–Sylvester formula) depending on badly-defined arguments (eigenvalues) we suggest a cumbersome function (hypergeometrical series) depending on well-defined arguments (matrix invariants). The function of matrix M ^ implies the sum of any formal finite or infinite power series of M ^ with arbitrary coefficients.