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

Abstract

We provide an explicit analytical evaluation of Ramanujan-type series for $1/\pi$. 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.

17 pages. Minor revision