Composita and its properties

Vladimir V. Kruchinin, Dmitry V. Kruchinin

Published 2011-03-14, updated 2013-03-25Version 2

In this paper we study the coefficients of the powers of an ordinary generating function and their properties. A new class of functions based on compositions of an integer $n$ is introduced and is termed composita. We present theorems about compositae and operations with compositae. We obtain the compositae of polynomials, trigonometric and hyperbolic functions. Using the notion of the composita we get the solution of the functional equation $B(x)=H(xB(x)^m)$, where $H(x),B(x)$ are generating functions, and $m\in \mathbb{N}$.