Commutation Relations, Normal Ordering, and Stirling Numbers

Abstract

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.