# Rational Approximations to A q-Analogue of π and Some Other q-Series

• Peter Bundschuh
Conference paper
Part of the Developments in Mathematics book series (DEVM, volume 16)

## Abstract

One of the famous mathematical constants is π, Archimedes’ constant. There are several analytic ways to define it, e.g., by the (slowly convergent) series
$$\pi = 4\sum\limits_{v = 0}^\infty {\frac{{\left( { - 1} \right)^v }} {{2^v + 1}},}$$
(1)
or by the (Gaussian probability density) integral
$$\pi = \left( {\int\limits_{ - \infty }^\infty {e^{ - x^2 } dx} } \right)^2 ;$$
(2)
for a comprehensive exposition of different representations and bibliography we refer the reader to [Fi, Section 1.4].

## Keywords

Irrationality q-analogues of mathematical constants basic hypergeometric series q-binomial theorem

## 2000 Mathematics subject classification

Primary 11J72 Secondary 11J82 33D15

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