The method for obtaining expressions for coefficients of reverse generating functions

TL;DR

This paper introduces a new class of functions called composita and proposes methods for deriving reverse and reciprocal generating functions, solving functional equations involving unknown generating functions.

Contribution

It presents a novel approach using composita to obtain expressions for reverse and reciprocal generating functions and solve related functional equations.

Findings

Defined the concept of composita for generating functions

Developed methods for calculating reciprocal and reverse functions

Provided solutions for functional equations involving unknown generating functions

Abstract

The powers of generating functions and its properties are analyzed. A new class of functions is introduced, based on the application of compositions of an integer $n$, called composita. The methods for obtaining reciprocal and reverse generating functions, and solutions of the functional equations $F(A(x))=G(x)$, where $A(x)$ is an unknown generating function, are proposed. Key words: generating functions, reverse, reciprocal, composita, method.