CMMSE: POWERS OF CATALAN GENERATING FUNCTIONS FOR BOUNDED OPERATORS

In this paper we consider the Catalan triangle numbers ( B n , k ) n ≥ 1 , 1 ≤ k ≤ n and ( A n , k ) n ≥ 1 , 1 ≤ k ≤ n + 1 to define powers of Catalan generating function C ( T ) where T is a linear and bounded operator on a Banach space X . When the operator 4 T is of power-bounded operator, the Catalan generating function is given by the Taylor series C ( T ) : = ∑ n = 0 ∞ C n T n , where c = ( C n ) n ≥ 0 is the Catalan sequence. Note that the operator C ( T ) is a solution of the quadratic equation T Y 2 − Y + I = 0 . We obtain new formulae which involves Catalan triangle numbers ( B n , k ) n ≥ 1 , 1 ≤ k ≤ n and ( A n , k ) n ≥ 1 , 1 ≤ k ≤ n + 1 . As element in the Banach algebra ℓ 1 ( N 0 , 1 4 n ) , we describe the spectrum of c ∗ j for j ≥1, and the expression of c −∗ j in terms of Catalan polynomials. In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future.