Series for $1/\pi$ Using Legendre's Relation

TL;DR

This paper introduces a novel approach to derive series for 1/π and other constants utilizing Legendre's relation, bypassing complex modular theory and connecting to elliptic functions and Legendre polynomials.

Contribution

The paper presents a new method based on Legendre's relation for generating series for 1/π, simplifying the process by avoiding modular theory and linking to elliptic functions.

Findings

Derived new series for 1/π using Legendre's relation

Connected Legendre polynomial values to Ramanujan series

Provided alternative generation functions for mathematical constants

Abstract

We present a new method for producing series for $1/\pi$ and other constants using Legendre's relation, starting from a generation function that can be factorised into two elliptic $K$'s; this way we avoid much of modular theory or creative telescoping. Many of our series involve special values of Legendre polynomials; their relationship to the more traditional Ramanujan series is discussed.