AN ANALYTICAL PROOF OF THE GENERAL COMPOSITION THEOREM FOR FORMAL POWER SERIES AND MATRIX REPRESENTATION

This paper further introduces the co-factors of the multinomial and uses them to reorganize the multinomial theorem. These co-factors will be applied to establish a relationship between the coefficients of fⁿ, the nth power of a unit formal power series f, and the coefficients of (f - f (0))ⁿ. Such a relationship yields the analytical proof of the general composition theorem for formal power series. This paper provides a matrix representation for the general composition of formal power series which generalizes the popular matrix representation for the composition of almost unit formal power series.