On a problem on Ramanujan-Sato series of level 11

Source abstract

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.