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arXiv · 2607.11924

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

Pablo Fernández Refolio
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Abstract

We provide an explicit analytical evaluation of Ramanujan-type series for $1/\pi$. 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.

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Pablo Fernández Refolio. 2026-07-09. Analytical Evaluation of Ramanujan Series for $1/\pi$ via Degree-2, Degree-3, Degree-7, and Degree-19 Modular Transformations. https://arxiv.org/abs/2607.11924

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