Convergence of Iterated Boolean-type Sums and Their Iterates

The convergence of iterated Boolean sums for different sequences of operators has been studied keeping fixed the number M of iterations or recently letting for a fixed operator. Here we consider some convergence properties of iterated Boolean-type sums of the classical Bernstein and Bernstein-Durrmeyer operators in the case where both the operators and their order of iterations are not fixed. In the case of Bernstein-Durrmeyer operators, we obtain a general solution of this problem. We can also state some alternative expressions of the semigroups and resolvent operators associated with the Voronovskaja's formula for these sequences of operators.