Analytical Evaluation of Ramanujan Series for $1/π$ via Degree-2, Degree-3, Degree-7, and Degree-19 Modular Transformations

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