A Comprehensive Treatment of q-Calculus

Addressing the persistently divergent views on notation that have hampered the development of q-calculus theory, this potent new notation method is based on logarithms. The book covers a multitude of q-notation topics and outlines its uses in modern physics.

To date, the theoretical development of q-calculus has rested on a non-uniform basis. Generally, the bulky Gasper-Rahman notation was used, but the published works on q-calculus looked different depending on where and by whom they were written. This confusion of tongues not only complicated the theoretical development but also contributed to q-calculus remaining a neglected mathematical field. This book overcomes these problems by introducing a new and interesting notation for q-calculus based on logarithms.For instance, q-hypergeometric functions are now visually clear and easy to trace back to their hypergeometric parents. With this new notation it is also easy to see the connection between q-hypergeometric functions and the q-gamma function, something that until now has been overlooked.

The book covers many topics on q-calculus, including special functions, combinatorics, and q-difference equations. Apart from a thorough review of the historical development of q-calculus, this book also presents the domains of modern physics for which q-calculus is applicable, such as particle physics and supersymmetry, to name just a few.

Covers many topics on q-calculus, i.e. special functions, combinatorics, q-difference equations and q-Bernoulli numbers Detailed coverage of the historical development of q-calculus Summarizes the domains of moderns physics for which q-calculus is applicable, i.e. particle physics and supersymmetry Introduction of a new logarithmic notation for q-calculus that supersedes the older Gaspar-Rahman notation Includes supplementary material: sn.pub/extras

Autorentext
Thomas Ernst, geboren 1974 in Mülheim an der Ruhr, zahlreiche literarische Veröffentlichungen und Drehbücher unter anderem für das ZDF. Lebt in Brüssel und arbeitet im Internationalen Literaturhaus "Passa Porta" sowie an der Universität Luxemburg an Projekten zum Thema "Sprache und Identität in Europa". www.thomasernst.net.

Klappentext

To date, the theoretical development of q-calculus has rested on a non-uniform basis. Generally, the bulky Gasper-Rahman notation was used, but the published works on q-calculus looked different depending on where and by whom they were written. This confusion of tongues not only complicated the theoretical development but also contributed to q-calculus remaining a neglected mathematical field. This book overcomes these problems by introducing a new and interesting notation for q-calculus based on logarithms. For instance, q-hypergeometric functions are now visually clear and easy to trace back to their hypergeometric parents. With this new notation it is also easy to see the connection between q-hypergeometric functions and the q-gamma function, something that until now has been overlooked.

The book covers many topics on q-calculus, including special functions, combinatorics, and q-difference equations. Beyond a thorough review of the historical development of q-calculus, it also presents the domains of modern physics for which q-calculus is applicable, such as particle physics and supersymmetry, to name just a few.

Inhalt
1 Introduction.- 2 The different languages of q.- 3 Pre q-Analysis.- 4 The q-umbral calculus and the semigroups. The Nørlund calculus of finite diff.- 5 q-Stirling numbers.- 6 The first q-functions.- 7 An umbral method for q-hypergeometric series.- 8 Applications of the umbral calculus.- 9 Ciglerian q-Laguerre polynomials.- 10 q-Jacobi polynomials.- 11 q-Legendre polynomials and Carlitz-AlSalam polynomials.- 12 q-functions of many variables.- 13 Linear partial q-difference equations.- 14 q-Calculus and physics.- 15 Appendix: Other philosophies.