# Commutation Relations, Normal Ordering, and Stirling Numbers

CRC Press, 18. sep. 2015 - 504 strani

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 = hVs 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.

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### Vsebina

 Chapter 1 Introduction 1 Chapter 2 Basic Tools 23 Chapter 3 Stirling and Bell Numbers 51 Chapter 4 Generalizations of Stirling Numbers 79 Chapter 5 Generalizations of Stirling Numbers Normal Ordering 139 Chapter 6 Normal Ordering in the Weyl AlgebraFurther Aspects 183 Chapter 7 The qDeformed Weyl Algebra and the Meromorphic Weyl Algebra 225 Chapter 8 A Generalization of the Weyl Algebra 277
 Appendix A Basic Definitions of qCalculus 399 Appendix B Symmetric Functions 401 Appendix C Basic Concepts in Graph Theory 403 Appendix D Definition and Basic Facts of Lie Algebras 405 Appendix E The BakerCampbellHausdorff Formula 409 Appendix F Hilbert Spaces and Linear Operators 411 Bibliography 419 Back Cover 481

 Chapter 9 The qDeformed Generalized Weyl Algebra 341 Chapter 10 A Generalization of Touchard Polynomials 375
 Avtorske pravice

### O avtorju (2015)

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.