Quadratic iterations to ${\pi }$ associated with elliptic functions to the cubic and septic base

Abstract

In this paper, properties of the functions A d (q), B d (q) and C d (q) are derived. Specializing at d = 1 and 2, we construct two new quadratic iterations to π. These are analogues of previous iterations discovered by the Borweins (1987), J. M. Borwein and F. G. Garvan (1997), and H. H. Chan (2002). Two new transformations of the hypergeometric series 2 F 1 (1/3, 1/6; 1; z) are also derived.