Generators of some Ramanujan formulas

TL;DR

This paper develops a novel method using WZ-pairs to generate Ramanujan-type formulas for 1/π without modular forms, introduces the concept of generators, and explores related binomial sums and convergence properties.

Contribution

It introduces the concept of generators of Ramanujan formulas using WZ-methods and demonstrates their use in deriving new series for 1/π and 1/π².

Findings

Successfully found some generators for Ramanujan-type series

Proved new binomial sums for 1/π²

Analyzed convergence rates of the series

Abstract

In this paper we prove some Ramanujan-type formulas for $1/\pi$ but without using the theory of modular forms. Instead we use the WZ-method created by H. Wilf and D. Zeilberger and find some hypergeometric functions in two variables which are second components of WZ-pairs that can be certified using Zeilberger's EKHAD package. These certificates have an additional property which allows us to get generalized Ramanujan-type series which are routinely proven by computer. We call these second hypergeometric components of the WZ-pairs generators. Finding generators seems a hard task but using a kind of experimental research (explained below), we have succeeded in finding some of them. Unfortunately we have not found yet generators for the most impressive Ramanujan's formulas. We also prove some interesting binomial sums for the constant $1/\pi^2$. Finally we rewrite many of the obtained series using pochhammer symbols and study the rate of convergence.