The different languages of q-calculus

Thomas Ernst
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Thomas Ernst: Uppsala University, Department of Mathematics

Chapter Chapter 2 in A Comprehensive Treatment of q-Calculus, 2012, pp 27-61 from Springer

Abstract: Abstract We give a survey of the different schools in q-analysis and introduce difference calculus and Bernoulli numbers to make a preparation for the important fourth chapter. 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, logarithms and calculus, because we claim that Fermat introduced the precursor of the q-integral long before calculus was invented. The Hindenburg combinatoric School gives a background to the discovery of the Schweins q-binomial theorem. The so-called Fakultäten was a forerunner to the Γ function and q-factorial. In the year 1844, Gudermann published his book on elliptic functions and two years later, in 1846, Heine published his important article on q-hypergeometric series, referring to Gauss’s Disquisitiones, pointing out the two q-analogues of the exponential function.

Keywords: Hypergeometric Function; Elliptic Function; Theta Function; Formal Power Series; Austrian School (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-0431-8_2

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DOI: 10.1007/978-3-0348-0431-8_2

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