A Comprehensive Treatment of q-Calculus

by Thomas Ernst

Overview

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.​

Product Details

ISBN-13:	9783034808064
Publisher:	Springer Basel
Publication date:	10/15/2014
Edition description:	2012
Pages:	492
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

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.​