Introduction

Thomas Ernst
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Thomas Ernst: Uppsala University, Department of Mathematics

Chapter Chapter 1 in A Comprehensive Treatment of q-Calculus, 2012, pp 1-25 from Springer

Abstract: Abstract In q-calculus we are looking for q-analogues of mathematical objects, which have the original object as limits when q tends to 1. In the q-analysis, we make formal proofs as follows. The validity of the obtained formula depends on the different parts of the proof. A typical proof in q-analysis looks as follows: Theorem ∗ $$ A=B \quad \mbox{\textit{or}}\quad A\cong{B}. $$ Here, ≅ means formal equality. We give a criterion for the validity of formula (∗). We introduce the fundamental concept of infinity (see Section 3.7 ) and make a comparison with nonstandard analysis. Throughout the whole book, we make a comparison with the units of physics to entice this important group of scientists. We make a complete list of analogies between the q-difference and q-sum operators and the differentiation or integration operator. We present the first q-functions in order to facilitate the description of the various schools.

Keywords: Theta Function; Formal Power Series; Summation Formula; Nonstandard Analysis; Reduction Formula (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_1

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DOI: 10.1007/978-3-0348-0431-8_1

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