Experimental Mathematics

Some New Formulas for 𝜋

Gert Almkvist, Christian Krattenthaler, and Joakim Petersson

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Abstract

We show how to find series expansions for {\small $\pi$} of the form {\small $\pi=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$}, where {\small $S(n)$} is some polynomial in n (depending on m, p, a). We prove that there exist such expansions for {\small $m=8k$, $p=4k$, $a=(-4)^k$}, for any k, and give explicit examples for such expansions for small values of m, p, a and a.

Article information

Source
Experiment. Math., Volume 12, Number 4 (2003), 441-456.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.em/1087568020

Mathematical Reviews number (MathSciNet)
MR2043994

Zentralblatt MATH identifier
1161.11419

Subjects
Primary: 11Y60: Evaluation of constants
Secondary: 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14]

Keywords
Fast converging series for Pi determinant evaluations

Citation

Almkvist, Gert; Krattenthaler, Christian; Petersson, Joakim. Some New Formulas for 𝜋. Experiment. Math. 12 (2003), no. 4, 441--456. https://projecteuclid.org/euclid.em/1087568020


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