  The powers of matrices with Stirling number-coefficients are investigated. It
is revealed that the elements of these matrices have a number of properties of
the ordinary Stirling numbers. Moreover, "higher order" Bell, Fubini and
Eulerian numbers can be defined. Hence we give a new interpretation for E. T.
Bell's iterated exponential integers. In addition, it is worth to note that
these numbers appear in combinatorial physics, in the problem of the normal
ordering of quantum field theoretical operators.
