The Reciprocal of sum_(ngeq 0)a^nb^n for non-commuting a and b, Catalan numbers and non-commutative quadratic equations

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.