Chapter

Diophantine Approximation

Volume 16 of the series Developments in Mathematics pp 123-139

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

  • Peter BundschuhAffiliated withMathematical Institute, University of Cologne
  • , Wadim ZudilinAffiliated withDepartment of Mechanics and Mathematics, Moscow Lomonosov State University

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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