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

  • Peter Bundschuh
  • Wadim Zudilin
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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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Peter Bundschuh
    • 1
  • Wadim Zudilin
    • 2
  1. 1.Mathematical InstituteUniversity of CologneCologneGermany
  2. 2.Department of Mechanics and MathematicsMoscow Lomonosov State UniversityMoscowRussia

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