Normally ordered forms of powers of differential operators and their combinatorics

Abstract

We investigate the combinatorics of the general formulas for the powers of the operator h partial derivative(d), where partial derivative is a differential operator on an arbitrary noncommutative ring in which h is central. New formulas for the generalized Stirling numbers are obtained, as well as results on the divisibility by primes of the coefficients which occur in the normally ordered form of h partial derivative(d). All of the above applies to the theory of formal differential operator rings.