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

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

Focusing on the singular moduli $k_{3}$,$k_{5}$,$k_{7}$,$k_{13}$ and $k_{37}$.

We demonstrate that the underlying elliptic identities can be established through lower-degree modular equations; specifically, we resolve the cases $m=3,7$ utilizing the modular equation of degree 2, the case $m=5$ via a combination of degree-2 and degree-3 modular equations, the case $m=13$ via a combination of degree-2 and degree-7 modular equations and the case $m=37$ via a combination of degree-2 and degree-19 modular equations.

In the first part of this work, we confine our attention to hypergeometric theory and elliptic integrals.

Subsequently, in the second part, we extend our framework to incorporate the theory of elliptic functions, which allows us to address the higher-complexity modular identities.