On powers of Stirling matrices

TL;DR

This paper explores the properties of matrices with Stirling number coefficients, introduces higher order combinatorial numbers, and connects these to applications in physics and quantum operator ordering.

Contribution

It provides new insights into Stirling matrices, defines higher order Bell, Fubini, and Eulerian numbers, and links them to combinatorial physics and quantum theory.

Findings

Properties of Stirling matrices are characterized.

Higher order Bell, Fubini, Eulerian numbers are defined.

Connections to quantum operator normal ordering are established.

Abstract

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.