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

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].