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.