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.

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.