Eisenstein Series, Alternative Modular Bases and Approximations of $1/\pi$

TL;DR

This paper uses Eisenstein series to evaluate hypergeometric functions and elliptic integrals, leading to a new Ramanujan-type formula for 1/π with high precision per term.

Contribution

It provides a complete evaluation of certain hypergeometric functions and derives a novel Ramanujan-type 1/π formula using Eisenstein series theory.

Findings

Explicit evaluations of hypergeometric functions in terms of elliptic integrals

Determination of the modulus of hypergeometric functions

A high-precision 1/π formula with 110 digits per term

Abstract

In this article using the theory of Eisenstein series, we give rise to the complete evaluation of two Gauss hypergeometric functions. Moreover we evaluate the modulus of each of these functions and the values of the functions in terms of the complete elliptic integral of the first kind. As application we give way of how to evaluate the parameters, in a closed-well posed form, of a general Ramanujan type $1/\pi$ formula. The result is a formula of 110 digits per term.