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

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

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Received: 18 February 2013; in revised form: 26 February 2013 / Accepted: 1 March 2013 / Published: 13 March 2013
Download PDF [301 KB, uploaded 13 March 2013]
Abstract: Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form k = 0 ( 1 2 ) k ( 1 d ) k ( d - 1 d ) k k ! 3 ( a k + 1 ) ( λ d ) k = δ π for d=2,3,4,6, where łd are singular values that correspond to elliptic curves with complex multiplication, and a,δ 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.
Keywords: Ramanujan type supercongruences; Atkin and Swinnerton-Dyer congruences; hypergeometric series; elliptic curves; complex multiplication; periods; modular forms; Picard–Fuchs equation Ramanujan type supercongruences; Atkin and Swinnerton-Dyer congruences; hypergeometric series; elliptic curves; complex multiplication; periods; modular forms; Picard–Fuchs equation
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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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.

AMA Style

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

Chicago/Turabian Style

Chisholm, Sarah; Deines, Alyson; Long, Ling; Nebe, Gabriele; Swisher, Holly. 2013. "ρ — Adic Analogues of Ramanujan Type Formulas for 1/π." Mathematics 1, no. 1: 9-30.


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