*Mathematical Methods in the Applied Sciences*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.

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