A family of Ramanujan-Orr formulas for $1/\pi$

TL;DR

This paper introduces a new variant of Wan's method to prove Ramanujan-Orr formulas for 1/π, simplifying the proof process by eliminating the need to solve systems of equations.

Contribution

It presents a novel variant of Wan's method that directly proves specific Ramanujan-Orr formulas for 1/π without solving systems of equations.

Findings

Proved two Ramanujan-Orr formulas for 1/π

Simplified the proof process for these formulas

Demonstrated the effectiveness of the new method

Abstract

We use a variant of Wan's method to prove two Ramanujan-Orr type formulas for $1/\pi$. This variant needs to know in advance the formulas for $1/\pi$ that we want to prove, but avoids the need of solving a system of equations.