2022/12/26 by John M. Campbell, Campbell, John M.
Mathematics · #33C20 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2212.13305
openalex publication_date 2022/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \frac1π 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 \frac1π 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.