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Mathematics 2013, 1(1), 9-30; doi:10.3390/math1010009

# ρ — Adic Analogues of Ramanujan Type Formulas for 1/π

1
Department of Mathematics and Statistics, University of Calgary, Calgary AB, T2N 1N4, Canada
2
Department of Mathematics, University of Washington, Seattle, WA 98195, USA
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Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
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Department of Mathematics, Iowa State University, Ames, IA 50011, USA
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Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
6
Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA
*
Author to whom correspondence should be addressed.
Received: 18 February 2013 / Revised: 26 February 2013 / Accepted: 1 March 2013 / Published: 13 March 2013

# Abstract

Following Ramanujan's work on modular equations and approximations of $\pi$, there are formulas for $1/\pi$ of the form Following Ramanujan's work on modular equations and approximations of $\pi$, there are formulas for $1/\pi$ of the form $\sum _{k=0}^{\infty }\frac{{\left(\frac{1}{2}\right)}_{k}{\left(\frac{1}{d}\right)}_{k}{\left(\frac{d-1}{d}\right)}_{k}}{k{!}^{3}}\left(ak+1\right){\left({\lambda }_{d}\right)}^{k}=\frac{\delta }{\pi }$ for $d=2,3,4,6,$ where ${ł}_{d}$ are singular values that correspond to elliptic curves with complex multiplication, and $a,\delta$ are explicit algebraic numbers. In this paper we prove a $p-$adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication. View Full-Text
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MDPI and ACS Style

Chisholm, S.; Deines, A.; Long, L.; Nebe, G.; Swisher, H. ρ — Adic Analogues of Ramanujan Type Formulas for 1/π. Mathematics 2013, 1, 9-30.

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