1st Edition
Commutation Relations, Normal Ordering, and Stirling Numbers
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.
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.
