Sundry topics
This chapter contains applications of the previous one. Examples are two quadratic q-hypergeometric transformations. Most of the summation formulas are given both in q-hypergeometric form and in q-binomial coefficient form for convenience. We use the q-an
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Introduction
1.1 A survey of the chapters Each section in the following chapters has an inbuilt chronology. The structure of the book is as follows: In Section 1.2 we will describe the general nature of the different equations in q-calculus together with the character of the proofs. In Section 1.3 we will briefly list the available books on special functions (and q-analysis) and give a survey of relations to complex analysis. Since the concept of infinity, ∞, is a central theme in q-analysis, we will briefly describe the relationship with nonstandard analysis in Section 1.4. We will make a comparison with physics concepts in Section 1.5, so that the reader is not lost in the many formulas. In Section 1.6 we will summarize the analogies between the q-difference and q-sum operators and the differentiation or integration operator in four tables. In Section 1.7 we will start with the first q-functions in order to facilitate the description of the various Schools. In the following sections we will sketch the connections to other subjects like analytic number theory and combinatorics. In Chapter 2 we will give a survey of the different Schools in q-analysis, with special emphasis on the Watson and Austrian Schools. In Section 2.3 we talk for the first time about difference calculus and Bernoulli numbers in order to make a preparation for the important fourth chapter. In Section 2.5 we summarize the different attempts at elliptic and theta functions, both of which are intimately related to q-calculus. We present the history of trigonometry, prosthaphaeresis and logarithms in Section 2.6 as a preparation for the introduction of q-complex numbers [175, 185], which were presented at the Conference in Honor of Allan Peterson, Novacella, Italy, in 2007. The development of calculus is treated in detail in Section 2.6, because we claim that Fermat introduced the precursor of the q-integral long before calculus was invented. We then continue with the historical development of symbols for derivative, formal equalities and finite sums. The reason is that we use different signs for definition, equality, formal equality and symbolic equality. J. Faulhaber’s discrete mathematics is presented briefly in Section 2.8. T. Ernst, A Comprehensive Treatment of q-Calculus, DOI 10.1007/978-3-0348-0431-8_1, © Springer Basel 2012
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Introduction
The Hindenburg combinatoric School, which is presented in Section 2.9, provided a basis for the discovery of the Schweins q-binomial theorem. In Section 2.10 we briefly describe the so-called Fakultäten, the forerunner of the function and q-factorial. We then come to the various German Schools of special functions that are described in Sections 2.11 to 2.17. In Section 2.18 we go through the transition from fluxions to the Leibniz notation in Cambridge, which enabled the umbral calculus and the works of Jackson and Cigler on q-calculus. In Sections 3.1 and 3.2 we present the duality between analytic number theory, combinatorial identities and q-series, to indicate the historical development o
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