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Analytical Evaluation of Ramanujan Series for 1/π 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/π. Focusing on the singular moduli kr 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.

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Cite

BibTeXRIS
BibTeX
@misc{refolio2026analytical,
  title = {{Analytical Evaluation of Ramanujan Series for $1/π$ via Degree-2, Degree-3, Degree-7, and Degree-19 Modular Transformations}},
  author = {Pablo Fernández Refolio},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2607.11924},
  eprint = {2607.11924},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.11924},
}
RIS
TY  - UNPB
TI  - Analytical Evaluation of Ramanujan Series for 1/π via Degree-2, Degree-3, Degree-7, and Degree-19 Modular Transformations
AU  - Pablo Fernández Refolio
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2607.11924
UR  - https://arxiv.org/abs/2607.11924
AB  - 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.
ER  -