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

Author

Listed:
  • Sarah Chisholm

    (Department of Mathematics and Statistics, University of Calgary, Calgary AB, T2N 1N4, Canada)

  • Alyson Deines

    (Department of Mathematics, University of Washington, Seattle, WA 98195, USA)

  • Ling Long

    (Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
    Department of Mathematics, Iowa State University, Ames, IA 50011, USA)

  • Gabriele Nebe

    (Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen, Germany)

  • Holly Swisher

    (Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA)

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 ł d 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.

Suggested Citation

  • Sarah Chisholm & Alyson Deines & Ling Long & Gabriele Nebe & Holly Swisher, 2013. "ρ — Adic Analogues of Ramanujan Type Formulas for 1/π," Mathematics, MDPI, vol. 1(1), pages 1-22, March.
    • Handle: RePEc:gam:jmathe:v:1:y:2013:i:1:p:9-30:d:24250
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