Analytical Evaluation of Ramanujan Series for 1/π via Degree-2, Degree-3, Degree-7, and Degree-19 Modular Transformations

2026/07/09 by Pablo Fernández Refolio · 2 voices
#math.GM

paper · pdf

Abstract

We provide an explicit analytical evaluation of Ramanujan-type series for 1/π. Focusing on the singular moduli k_r for r ∈ \5, 7, 13, 37\, we demonstrate that the underlying elliptic identities can be established through lower-degree modular transformations. Specifically, we resolve the case r = 5 via a combination of degree-2 and degree-3 modular transformations; the case r = 7 utilizing the modular transformation of degree 2; the case r = 13 via a combination of degree-2 and degree-7 modular transformations; and the case r = 37 via a combination of degree-2 and degree-19 modular transformations.

Discussions

• We will show the mentioned case related to $\alpha(37)$ . The complete elliptic integrals satisfy Landen descending transformations \begin{align}\label{eq:landendownwardke} K(\frac{2\sqrt{k}}{1+k})=K( [stackexchange, 2 points]
• Let $\tau$ denote a complex number with positive imaginary part. We define the nome $q(\tau) $ (also denoted by $q_\tau$ or just $q$ when $\tau$ is available from context) as $q(\tau) =\exp (\pi i\tau [stackexchange, 1 points]

Related

• Ramanujan-Type Series of Signature 2: Analytical Evaluation via Degree-2 Modular Transformations
• The Explicit Evaluations Formula for Ramanujan's Singular Moduli and Ramanujan- Selberg Continued Fraction
• P--Q "mixed" modular equations of degree 15
• Kind of proofs of Ramanujan-like series
• A Matrix form of Ramanujan-type series for 1/π