# Commutation Relations, Normal Ordering, and Stirling Numbers

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## Book Description

**Commutation Relations, Normal Ordering, and Stirling Numbers** provides an introduction to the combinatorial aspects of normal ordering in the Weyl algebra and some of its close relatives. The Weyl algebra is the algebra generated by two letters *U* and *V* subject to the commutation relation *UV − VU = I*. It is a classical result that normal ordering powers of *VU* involve the Stirling numbers.

The book is a one-stop reference on the research activities and known results of normal ordering and Stirling numbers. It discusses the Stirling numbers, closely related generalizations, and their role as normal ordering coefficients in the Weyl algebra. The book also considers several relatives of this algebra, all of which are special cases of the algebra in which *UV − qVU = hV ^{s}* holds true. The authors describe combinatorial aspects of these algebras and the normal ordering process in them. In particular, they define associated generalized Stirling numbers as normal ordering coefficients in analogy to the classical Stirling numbers. In addition to the combinatorial aspects, the book presents the relation to operational calculus, describes the physical motivation for ordering words in the Weyl algebra arising from quantum theory, and covers some physical applications.

## Table of Contents

**Introduction **Set Partitions, Stirling, and Bell Numbers

Commutation Relations and Operator Ordering

Normal Ordering in the Weyl Algebra and Relatives

Content of the Book

**Basic Tools**Sequences

Solving Recurrence Relations

Generating Functions

Combinatorial Structures

Riordan Arrays and Sheffer Sequences

**Stirling and Bell Numbers **Definition and Basic Properties of Stirling and Bell Numbers

Further Properties of Bell Numbers

*q*-Deformed Stirling and Bell Numbers

(

*p, q*)-Deformed Stirling and Bell Numbers

**Generalizations of Stirling Numbers **Generalized Stirling Numbers as Expansion Coefficients in Operational Relations

Stirling Numbers of Hsu and Shiue: A Grand Unification

Deformations of Stirling Numbers of Hsu and Shiue

Other Generalizations of Stirling Numbers

**The Weyl Algebra, Quantum Theory, and Normal Ordering**

The Weyl Algebra

Short Introduction to Elementary Quantum Mechanics

Physical Aspects of Normal Ordering

**Normal Ordering in the Weyl Algebra—Further Aspects**

Normal Ordering in the Weyl Algebra

Wick’s Theorem

The Monomiality Principle

Further Connections to Combinatorial Structures

A Collection of Operator Ordering Schemes

The Multi-Mode Case

**The q-Deformed Weyl Algebra and the Meromorphic Weyl Algebra**

Remarks on

*q*-Commuting Variables

The

*q*-Deformed Weyl Algebra

The Meromorphic Weyl Algebra

The

*q*-Meromorphic Weyl Algebra

**A Generalization of the Weyl Algebra**

Definition and Literature

Normal Ordering in Special Ore Extensions

Basic Observations for the Generalized Weyl Algebra

Aspects of Normal Ordering

Associated Stirling and Bell Numbers

**The q-Deformed Generalized Weyl Algebra **Definition and Literature

Basic Observations

Binomial Formula

Associated Stirling and Bell Numbers

**A Generalization of Touchard Polynomials **Touchard Polynomials of Arbitrary Integer Order

Outlook: Touchard Functions of Real Order

Outlook: ComtetTouchard Functions

Outlook:

*q*-Deformed Generalized Touchard Polynomials

Appendices

Bibliography

Indices

*Exercises appear at the end of each chapter.*

## Author(s)

### Biography

**Toufik Mansour** is a professor at the University of Haifa. His research interests include enumerative combinatorics and discrete mathematics and its applications. He has authored or co-authored numerous papers in these areas, many of them concerning the enumeration of normal ordering. He earned a PhD in mathematics from the University of Haifa.

**Matthias Schork** is a member of the IT department at Deutsche Bahn, the largest German railway company. His research interests include mathematical physics as well as discrete mathematics and its applications to physics. He has authored or coauthored many papers focusing on Stirling numbers and normal ordering and its ramifications. He earned a PhD in mathematics from the Johann Wolfgang Goethe University of Frankfurt.