The Reciprocal of $\sum_{n\geq 0}a^nb^n$ for non-commuting $a$ and $b$, Catalan numbers and non-commutative quadratic equations

TL;DR

This paper investigates the inverse of a non-commutative power series related to Catalan numbers, revealing it satisfies a quadratic equation and exploring solutions to similar equations.

Contribution

It provides a formal series expression for the inverse of a non-commutative sum and links it to Catalan numbers, introducing new insights into non-commutative quadratic equations.

Findings

Inverse series satisfies a non-commutative quadratic equation

Number of specific monomials equals Catalan number

General solutions to similar quadratic equations are studied

Abstract

The aim of this paper is to describe the inversion of the sum $\sum_{n\geq 0}a^nb^n$ where $a$ and $b$ are non-commuting variables as a formal series in $a$ and $b$. We show that the inversion satisfies a non-commutative quadratic equation and that the number of certain monomials in its homogeneous components equals to a Catalan number. We also study general solutions of similar quadratic equations.