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