New Generalizations of Two Ramanujan Series for 1/π
Abstract
By utilizing two known hypergeometric summation identities, we extend two \mbox{well-known} Ramanujan series for 1/π by making each of them a member of an infinite family of series. We also find the corresponding families involving harmonic numbers and odd harmonic numbers. To further illustrate our method we derive families of \mbox{Ramanujan-like} series for 1/π associated with a hypergeometric series derived by Lavoie, two hypergeometric series from Bailey's book and a recent hypergeometric series found by Campbell. Thus, by leveraging known hypergeometric summation identities, our approach bypasses the more tedious, traditional limits or complex Fourier-Legendre expansions; offering a cleaner, unified template for generating Ramanujan-like series for 1/π.
Related
- Gosper's and Ramanujan's infinite products and generalizations
- Generalized Glaisher-Kinkelin constants and Ramanujan summation of series
- Ramanujan-Type Series of Signature 2: Analytical Evaluation via Degree-2 Transformations and Associated Harmonic Expansions
- Analytical Evaluation of Ramanujan Series for 1/π via Degree-2, Degree-3, Degree-7, and Degree-19 Modular Transformations
- Borwein-Bradley-type identities for Ramanujan-type series for 1/π4
BibTeX
@misc{adegoke2026generalizations,
title = {{New Generalizations of Two Ramanujan Series for $1/π$}},
author = {Kunle Adegoke},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09209},
eprint = {2609.09209},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09209},
}
RIS
TY - UNPB
TI - New Generalizations of Two Ramanujan Series for 1/π
AU - Kunle Adegoke
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.09209
UR - https://arxiv.org/abs/2609.09209
AB - By utilizing two known hypergeometric summation identities, we extend two \mbox{well-known} Ramanujan series for 1/π by making each of them a member of an infinite family of series. We also find the corresponding families involving harmonic numbers and odd harmonic numbers. To further illustrate our method we derive families of \mbox{Ramanujan-like} series for 1/π associated with a hypergeometric series derived by Lavoie, two hypergeometric series from Bailey's book and a recent hypergeometric series found by Campbell. Thus, by leveraging known hypergeometric summation identities, our approach bypasses the more tedious, traditional limits or complex Fourier-Legendre expansions; offering a cleaner, unified template for generating Ramanujan-like series for 1/π.
ER -