Analytical Evaluation of Ramanujan Series for $1/π$ via Degree-2, Degree-3, Degree-7, and Degree-19 Modular Transformations
Year: 2026
Type: preprint
Abstract: We provide an explicit analytical evaluation of Ramanujan-type series for $1/π$. Focusing on the singular moduli $k_{r}$ for $r \in \{5, 7, 13, 37\}$, 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.
Source: arXiv (Cornell University)
Author: Refolio, Pablo Fernández
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7168354229