This material has been published in
Experiment. Math. 12 (2003),
441-456,
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Gert Almkvist, Christian Krattenthaler and Joakim Petersson
Some new formulas for pi
(28 pages)
Abstract.
We show how to find arbitrarily fast convergent
series expansions for \pi of the
form
\pi=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}, where
S(n) is some polynomial in n (depending on
m,p,a).
We prove that there exist such
expansions for m=8k, p=4k,
a=(-4)k, for any k, and give
explicit examples for such expansions for small values of m,
p and a.
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