发表机构
Dalhousie University（达尔豪斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了关于第11级Ramanujan-Sato级数系数$c_{11}(n)$的长期未解问题，通过引入并证明一个双二项式求和公式，直接类比较低级系数公式，成功给出了所有非负整数$n$的显式表达式。
AI 中文摘要
一个关于Ramanujan-Sato级数的长期未解问题，是确定与第11级Ramanujan-Sato级数相关联的系数$c_{11}(n)$的表达式，该表达式对所有非负整数$n$具有固定的、嵌套的有限二项式求和，直接类比于较低级Ramanujan-Sato级数的已知系数公式（这一点在我们的论文中已澄清并明确说明）。该未解问题由Cooper等人于2015年[J. Math. Anal. Appl.]、Cooper 2017年关于Ramanujan theta函数的教科书以及Cooper 2025年对Contemp. Math.系列的贡献中提出。此问题似乎一直未解。我们通过引入并证明$c_{11}(n)$的双二项式求和公式成功解决了此问题，该公式直接类似于较低级Ramanujan-Sato级数的系数公式。
英文摘要
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.
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