Quadratic irrational analogues of Ramanujan's series for $1/\pi$

Abstract

About 40 years ago Jonathan and Peter Borwein discovered the series identity $$ \sum_{n=0}^\infty \frac{(-1)^n(6n)!}{(3n)!(n!)^3} \frac{(A+nB)}{C^{n+1/2}} = \frac{1}{12\pi} $$ where \begin{align*} A&=1657145277365+212175710912\sqrt{61},\cr B&=107578229802750+13773980892672\sqrt{61},\cr C&=\left(5280(236674+30303\sqrt{61})\right)^3 \end{align*} which adds roughly 25 digits of accuracy per term. They noted that if each of the quadratic irrationals $A$, $B$ and $C$ is replaced by their conjugates, that is, each number $a+b\sqrt{61}$ is changed to $a-b\sqrt{61}$, then the resulting series also converges to a rational multiple of $1/\pi$. They gave several other examples of quadratic irrational series for $1/\pi$, and noted that the conjugate series converges to another rational multiple of $1/\pi$ or in some cases the conjugate series diverges. The purpose of this work is to provide an explanation and classification of such series. Our classification includes Ramanujan's 17 original series, as well as series of the Borweins, Chudnovskys, Sato and others. We extend the classification to genus-zero subgroups $\Gamma_0(\ell)+$, that is, for each $\ell \in \big\{1,2,3,\ldots,36,38,39,41,42,44,45,46,47,49,50,51,54,55,56,59,60,62,66,69,70, 71,78,87,92,94,95,105,110,119\big\}$ we calculate the Hauptmoduls, associated weight two modular forms, and the corresponding rational and real quadratic irrational series for $1/\pi$. The classification reveals many interrelations among the different series. For example, we show that the Borweins' series above, and its conjugate, are equivalent by hypergeometric transformation formulas to the level~7 rational series $$ \sum_{n=0}^\infty \left\{\sum_{j=0}^n {n \choose j}^2{2j \choose n} {n+j \choose j}\right\} (11895n+1286) \frac{(-1)^n}{22^{3n+3}} = \frac{1}{\pi\sqrt{7}}. $$