Variations on a theme of Ap\'ery

TL;DR

This paper explores modifications of Apéry's continued fractions, leading to new rapidly converging continued fractions for mathematical constants, expanding on Apéry's original discoveries and their applications.

Contribution

It introduces novel modifications to Apéry's continued fractions, resulting in new rapidly converging representations for specific mathematical constants.

Findings

New continued fractions with rapid convergence for certain constants

Extensions of Apéry's methods to other constants

Potential applications in computational mathematics

Abstract

Ap\'ery's remarkable discovery of rapidly converging continued fractions with small coefficients for $\zeta(2)$ and $\zeta(3)$ has led to a flurry of important activity in an incredible variety of different directions. Our purpose is to show that modifications of Ap\'ery's continued fractions can give interesting results including new rapidly convergent continued fractions for certain interesting constants.