Composita and its properties

TL;DR

This paper introduces the concept of composita, a new class of functions based on compositions of integers, and explores their properties, operations, and applications to solving functional equations involving generating functions.

Contribution

The paper defines composita, develops theorems about their properties and operations, and applies them to derive compositae of various functions and solve specific functional equations.

Findings

Derived compositae of polynomials, trigonometric, and hyperbolic functions.

Provided methods to compute compositae for new classes of functions.

Solved a functional equation involving generating functions using composita.

Abstract

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 $m\in \mathbb{N}$.