On a problem on Ramanujan-Sato series of level 11
Abstract
A longstanding open problem concerning Ramanujan-Sato series is given by determining an expression for the coefficients c11(n) associated with Ramanujan-Sato series of level 11 with a fixed, nested, finite binomial sum for all nonnegative integers n, by direct analogy with the known formulas for the coefficients for lower-level Ramanujan-Sato series (and this is clarified and made explicit in our paper). This open problem was given by Cooper et al. in 2015 [J. Math. Anal. Appl.], in Cooper's 2017 textbook on Ramanujan's theta functions, and in Cooper's 2025 contribution to the Contemp. Math. series. This problem seems to have remained open. We succeed in solving this problem, by introducing and proving a double binomial sum formula for c11(n) that is directly analogous to the coefficient formulas for lower-level Ramanujan-Sato series.
Related
- 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
- New Generalizations of Two Ramanujan Series for 1/π
- Borwein-Bradley-type identities for Ramanujan-type series for 1/π4
BibTeX
@misc{campbell2026problem,
title = {{On a problem on Ramanujan-Sato series of level 11}},
author = {John M. Campbell},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00551},
eprint = {2610.00551},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00551},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a problem on Ramanujan-Sato series of level 11
AU - John M. Campbell
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00551
UR - https://arxiv.org/abs/2610.00551
AB - A longstanding open problem concerning Ramanujan-Sato series is given by determining an expression for the coefficients c_11(n) associated with Ramanujan-Sato series of level 11 with a fixed, nested, finite binomial sum for all nonnegative integers n, by direct analogy with the known formulas for the coefficients for lower-level Ramanujan-Sato series (and this is clarified and made explicit in our paper). This open problem was given by Cooper et al. in 2015 [J. Math. Anal. Appl.], in Cooper's 2017 textbook on Ramanujan's theta functions, and in Cooper's 2025 contribution to the Contemp. Math. series. This problem seems to have remained open. We succeed in solving this problem, by introducing and proving a double binomial sum formula for c_11(n) that is directly analogous to the coefficient formulas for lower-level Ramanujan-Sato series.
ER -