Time- and Space-Efficient Evaluation of Some Hypergeometric Constants

Abstract

The currently best known algorithms for the numerical evaluation of hypergeometric constants such as $\hat{I}\P (3)$ to $d$ decimal digits have time complexity $O(M(d){\log }^{2}d)$ and space complexity of $O(d\log d)$ or $O(d)$. Following work from Cheng, Gergel, Kim and Zima, we present a new algorithm with the same asymptotic complexity, but more efficient in practice. Our implementation of this algorithm improves slightly over existing programs for the computation of $\ddot{I}€$, and we announce a new record of 2 billion digits for $\hat{I}\P (3)$.