An extension of the Chudnovsky algorithm

Abstract

Using an infinite family of generalizations of the Chudnovsky brothers' series recently obtained via the analytic continuation of the Borwein brothers' formula for Ramanujan-type series of level 1, we apply the Gauss-Salamin-Brent iteration for $π$ to obtain a new, Ramanujan-type series that yields more digits per term relative to current world record given by an extension of the Chudnovsky algorithm from Bagis and Glasser that produces about 110 digits per term. We explicitly evaluate the required nested radicals over $\mathbb{Q}$ involved in the our summation, which yields about 153 digits per term, and we provide a practical way of implementing our higher-order version of the Chudnovsky algorithm via the the PARI/GP system. An evaluation due to Berndt and Chan for the modular $j$-invariant associated with their order-3315 extension of the Chudnovskys' Ramanujan-type series provides a key to our applications of recursions for the elliptic lambda and elliptic alpha functions.