Abstract

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.

Keywords and Phrases

Formal power series, power series, composition, matrix representation, multinomial theorem.

A.M.S. subject classification

Primary: 13F25; Secondary: 40A05, 15B33.

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