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Apéry Acceleration of Continued Fractions

Henri Cohen

Abstract

We explain in detail how to accelerate continued fractions (for constants as well as for functions) using the method used by R.~Apéry in his proof of the irrationality of $ζ(3)$. We show in particular that this can be applied to a large number of continued fractions which can be found in the literature, thus providing a large number of new continued fractions. As examples, we give a new continued fraction for $\log(2)$ and for $ζ(3)$, as well as a simple proof of one due to Ramanujan.

Apéry Acceleration of Continued Fractions

Abstract

We explain in detail how to accelerate continued fractions (for constants as well as for functions) using the method used by R.~Apéry in his proof of the irrationality of . We show in particular that this can be applied to a large number of continued fractions which can be found in the literature, thus providing a large number of new continued fractions. As examples, we give a new continued fraction for and for , as well as a simple proof of one due to Ramanujan.
Paper Structure (14 sections, 47 equations)

This paper contains 14 sections, 47 equations.

Table of Contents

  1. Introduction and Notation
  2. Bauer--Muir Acceleration
  3. Apéry's method
  4. Basic Examples
  5. $\log(2)$ and $\psi(z+1/2)-\psi(z)$
  6. $\zeta(2)=\pi^2/6$ and $\psi'(z)$
  7. $\zeta(3)$ and $\psi"(z)$
  8. An Example where Apéry can be Iterated
  9. Generalizations of Apéry's Method
  10. Case when $r(n,l)$ is a rational function in $l$
  11. Need for a simplifier
  12. Need for a multiplier: the Case of $\zeta(4)=\sum_{n\ge1}1/n^4$
  13. Need for a multiplier: a New CF for $\zeta(3)$
  14. Slowing Down Apéry's Method