A General Method for Constructing Ramanujan Formulas for $1/\pi^{\nu}$

TL;DR

This paper presents a theoretical framework for generating Ramanujan-type formulas for powers of 1/π, including explicit constructions for 1/π^4 and 1/π^6, utilizing elliptic functions.

Contribution

It introduces a general method for constructing Ramanujan formulas for 1/π^{2ν} and provides explicit series for 1/π^4 and 1/π^6.

Findings

Constructed series for 1/π^4 and 1/π^6

Developed a theoretical background for Ramanujan-type formulas

Studied elliptic alpha function for evaluations

Abstract

In this article we give the theoretical background for generating Ramanujan type $1/\pi^{2\nu}$ formulas. As applications of our method we give a general construction of $1/\pi^4$ series and examples of $1/\pi^6$ series. We also study the elliptic alpha function whose values are useful for such evaluations.