Extensions of Ramanujan's two formulas for $1/\pi$

TL;DR

This paper extends two of Ramanujan's formulas for 1/π using hypergeometric methods and derives five additional parametrized summation formulas for 1/π.

Contribution

It introduces new extended formulas for 1/π based on hypergeometric techniques and provides five parametrized summation formulas, expanding Ramanujan's original results.

Findings

Extended Ramanujan's formulas for 1/π

Derived five new parametrized summation formulas

Used hypergeometric method for derivations

Abstract

In terms of the hypergeometric method, we establish the extensions of two formulas for $1/\pi$ due to Ramanujan [27]. Further, other five summation formulas for $1/\pi$ with free parameters are also derived in the same way.