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

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.

Elliptic Functions; Pi