Powers of Catalan generating functions for bounded operators

Abstract

Let be the Catalan sequence and a linear and bounded operator on a Banach space such is a power‐bounded operator. The Catalan generating function is defined by the following Taylor series: Note that the operator is a solution of the quadratic equation . In this paper, we define powers of the Catalan generating function in terms of the Catalan triangle numbers. We obtain new formulae that involve Catalan triangle numbers: the spectrum of and the expression of for in terms of Catalan polynomials ( is the usual convolution product in sequences). In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future.