Hypergeometric accelerations with shifted indices

TL;DR

This paper extends hypergeometric acceleration techniques using shifted indices to derive new and existing series for constants like 1/π, building on prior work by Chu, Zhang, and Wilf.

Contribution

It introduces a novel acceleration method with shifted indices for hypergeometric series, recovering and generating many series for mathematical constants.

Findings

Recovered many known accelerated series for constants

Generated new inequivalent series involving Ramanujan-type formulas

Extended hypergeometric acceleration techniques with shifted indices

Abstract

Chu and Zhang, in 2014, introduced hypergeometric transforms derived through the application of an Abel-type summation lemma to Dougall's ${}_{5}H_{5}$-series. These transforms were applied by Chu and Zhang to obtain accelerated rates of convergence, yielding rational series related to the work of Ramanujan and Guillera. We apply a variant of an acceleration method due to Wilf using what we refer to as shifted indices for Pochhammer symbols involved in our first-order, inhomogeneous recurrences derived via Zeilberger's algorithm, to build upon Chu and Zhang's accelerations, recovering many of their accelerated series and introducing many inequivalent series for universal constants, including series of Ramanujan type involving linear polynomials as summand factors, as in Ramanujan's series for $\frac{1}{\pi}$.