On a Ramanujan-type series associated with the Heegner number 163

Abstract

Using the Wolfram NumberTheory package and the Recognize command, together with numerical estimates involving the elliptic lambda and elliptic alpha functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type series related to the class number $h(-d)=1$ for a quadratic form with discriminant $d=163$. This conjectured series is of level one and has positive terms, and recalls the Chudnovsky brothers' alternating series of the same level, given the connection between the Chudnovsky-Chudnovsky formula and the Heegner number $d=163$ such that $\mathbb{Q}\left(\sqrt{-d}\right)$ has class number one. We prove Bagis and Glasser's conjecture by proving evaluations for ${\lambda }^{\ast }(163)$ and $\alpha (163)$, which we derive using the Chudnovsky brothers' formula together with the analytic continuation of a formula due to the Borwein brothers for Ramanujan-type series of level one. As a byproduct of our method, we obtain an infinite family of Ramanujan-type series for $\frac{1}{\pi }$ generalizing the Chudnovsky algorithm.

To appear in the Journal of Number Theory