AI 中文总结
该研究对\(1/\pi\)的拉马努金级数进行解析评估，聚焦奇异模，通过低次模方程建立椭圆恒等式，先基于超几何理论和椭圆积分，后扩展到椭圆函数理论，以处理更高复杂度的模恒等式。
AI 中文摘要
我们对\(1/\pi\)的拉马努金型级数进行了明确的解析评估。聚焦于奇异模\(k_{3}\)、\(k_{5}\)、\(k_{7}\)、\(k_{13}\)和\(k_{37}\)。证明了相关椭圆恒等式可通过低次模方程建立，如利用二次模方程解决\(m = 3,7\)的情况，通过二次和三次模方程组合解决\(m = 5\)的情况等。第一部分关注超几何理论和椭圆积分，第二部分扩展框架纳入椭圆函数理论以处理更高复杂度的模恒等式。
英文摘要
We provide an explicit analytical evaluation of Ramanujan-type series for $1/π$. Focusing on the singular moduli $k_{r}$ for $r \in \{5, 7, 13, 37\}$, 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.
Comments17 pages. Minor revision
