doi:10.1007/s43034-023-00290-0engMiana, Pedro J.Romero, NataliaCatalan generating functions for bounded operatorsART-2023-134550In this paper, we study the solution of the quadratic equation TY2−Y+I=0 where T is a linear and bounded operator on a Banach space X. We describe the spectrum set and the resolvent operator of Y in terms of the ones of T. In the case that 4T is a power-bounded operator, we show that a solution (named Catalan generating function) of the above equation is given by the Taylor series C(T):=∑n=0∞CnTn, where the sequence (Cn)nis the well-known Catalan numbers sequence. We express C(T) by means of an integral representation which involves the resolvent operator (λT)−1. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices T which involves Catalan numbers2023http://zaguan.unizar.es/record/12755910.1007/s43034-023-00290-0http://zaguan.unizar.es/record/127559oai:zaguan.unizar.es:127559Annals of functional analysis 14, 69 (2023), 21byhttp://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccess