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

TL;DR

This work provides a unified, modular-framework classification of Ramanujan-type series for $1/\pi$ that involve quadratic irrationals. By attaching a Hauptmodul $X$ and a weight-two modular form $Z$ to each genus-zero $\Gamma_0(\ell)^+$, and using polynomials $G,H$ to encode a differential equation and a recurrence for the $Z$-expansion coefficients, the authors derive systematic conditions for when a quadratic irrational series converges to a multiple of $1/\pi$ and when it is equivalent to a rational or another quadratic irrational instance. The theory recovers Ramanujan’s original 17 series, the Borweins’, Chudnovskys’, and Sato’s results, and extends to a comprehensive catalog across many levels, including 36 rational series and numerous quadratic irrational cases, together with explicit hypergeometric transformations linking different levels. The work also clarifies when conjugate (quadratic) series diverge or map to conjugate-rational counterparts and provides extensive tables of Hauptmoduls, associated modular forms, and recurrence data, enabling precise computation and cross-checks. Overall, the paper deepens understanding of how modular-form structure governs Ramanujan-type $1/\pi$ series and exposes rich interconnections across levels via hypergeometric and CM-point methods, with practical impact for both theory and high-precision computations.

Abstract

About 40 years ago Jonathan and Peter Borwein discovered the series identity $$ \sum_{n=0}^\infty \frac{(-1)^n(6n)!}{(3n)!(n!)^3} \frac{(A+nB)}{C^{n+1/2}} = \frac{1}{12π} $$ 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 $1/π$. They gave several other examples of quadratic irrational series for $1/π$, and noted that the conjugate series converges to another rational multiple of $1/π$ 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 $Γ_0(\ell)+$, 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 $1/π$. 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 $$ \sum_{n=0}^\infty \left\{\sum_{j=0}^n {n \choose j}^2{2j \choose n} {n+j \choose j}\right\} (11895n+1286) \frac{(-1)^n}{22^{3n+3}} = \frac{1}{π\sqrt{7}}. $$

Figures (1)

• Figure 1: Maple code to compute the expansions of $X$ and $Z$, given $G$ and $H$.

Theorems & Definitions (24)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 6.1
• proof