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A Comprehensive Treatment of q-Calculus pp 1–25Cite as

Birkhäuser

Introduction

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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.

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Authors and Affiliations

  1. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Thomas Ernst

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Ernst, T. (2012). Introduction. In: A Comprehensive Treatment of q-Calculus. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0431-8_1

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