Variations on a theme of Apéry

TL;DR

This work investigates variations of Apéry's rapidly converging continued fractions, focusing on half-integer shifts and continuous parameter generalisations to generate new, elegant CFs for fundamental constants such as $\zeta(2)$, $\zeta(3)$ and $\log(2)$. By combining Apéry acceleration with Beukers-type integrals, hypergeometric and Meijer-G representations, and modular-form period theory, the authors derive new recurrences, closed-form gamma-quotient limits, and connections to Bessel moments and L-values, while organizing the arithmetic structure via Rhin–Viola groups. Although no new irrationality results are claimed, the paper provides a coherent framework linking CFs, special values, and periods, and highlights how half-shifts and parameter continuations can yield rich, computable identities and potential arithmetic consequences. The results thus offer a versatile toolkit for generating and understanding rapidly convergent CFs across constants, with deep ties to hypergeometric integrals, modular forms, and gamma quotients.

Abstract

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

Theorems & Definitions (18)

• Theorem 1.1
• proof
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof