\(p\)-adic analogues of Ramanujan type formulas for \(1/\pi\)
Summary: Following Ramanujan's work on modular equations and approximations of \(\pi\), there are formulas for \(1/\pi\) of the form \[ \sum\limits_{k=0}^\infty\frac{(\frac{1}{2})k(\frac{1}{d})k(\frac{d-1}{d})k}{k!^3}(ak+1)(\lambda_d)^k=\frac{\delta}{\pi} \] for \(d=2,3,4,6\), where \(\lambda d\) 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.
scientific article; zbMATH DE number 6334773
scientific article