Applications of a class of transformations of complex sequences

TL;DR

This paper leverages Mishev's 2018 sequence transformation to simplify proofs of Ramanujan's series for 1/π, extend related hypergeometric identities, and generalize Ramanujan-like series involving binomial coefficients and harmonic numbers.

Contribution

It introduces a novel application of Mishev's transform to provide simplified proofs and generalizations of Ramanujan-type series for 1/π and related hypergeometric identities.

Findings

Simplified proof of Ramanujan's series for 1/π.

Extended hypergeometric identities using Mishev's transform.

Generalized Ramanujan-like series involving binomial coefficients and harmonic numbers.

Abstract

Through an application of a remarkable result due to Mishev in 2018 concerning the inverses for a class of transformations of sequences of complex numbers, we obtain a very simple proof for a famous series for $\frac{1}{\pi}$ due to Ramanujan. We then apply Mishev's transform to provide proofs for a number of related hypergeometric identities, including a new and simplified proof for a family of series for $\frac{1}{\pi}$ previously obtained by Levrie via Fourier--Legendre theory. We generalize this result using Mishev's transform, so as to extend a result due to Guillera on a Ramanujan-like series involving cubed binomial coefficients and harmonic numbers.