Finding the continued fraction for the "tails" of $\eta(3)$

Question

I am interested in the continued fractions for the "tails" or "correction term" of the series sum of specific constants. For example, the Madhava's correction term for $\pi/4$:

$$ \frac{\pi}{4}=\sum_{k=1}^{n}\frac{(-1)^{k-1}}{2k-1}+\frac{(-1)^n}{2}\cdot \frac{1}{2n+{\overset{\infty }{\underset{k=1}{\Large\vcenter{\hbox{K}}}}~\dfrac{k^2}{2n}}}$$

and the "tails" for $\zeta(2)$:

$$ \zeta(2)=\sum_{k=1}^{n}\frac{1}{k^2}+\frac{2}{2n+1+{\overset{\infty }{\underset{k=1}{\Large\vcenter{\hbox{K}}}}~\dfrac{k^4}{(2k+1)(2n+1)}}}$$

This post from Wolfgang shows the continued fraction "tails" for $\zeta(3)$, also discusses the "tails" for $\eta(2)$ and $\beta(2)$ (Catalan's constant). The comment of the post from Henri Cohen indicates that a paper (STDN 1979-1980 by Batut-Olivier, Expose 23) had studied this subject and revealed several continued fraction "tails" for the constants like $\pi/4, \ln(2), \eta(2), \zeta(2), \beta(2)$ and $\zeta(3)$. This paper developed a general method to derive such continued fraction "tails" for a series sum. But it's in French and I can't figure out the process how the author derived these continued fractions.

I am trying to find out the continued fraction "tails" for $\eta(3)$ without success. From known facts:

$$ \ln(2)=\eta(1)=\sum_{k=1}^{n}\frac{(-1)^{k-1}}{k}+(-1)^n\cdot \frac{1}{2n+1+{\overset{\infty }{\underset{k=1}{\Large\vcenter{\hbox{K}}}}~\dfrac{k^2}{2n+1}}}$$

$$ \eta(2)=\sum_{k=1}^{n}\frac{(-1)^{k-1}}{k^2}+\frac{(-1)^n}{2}\cdot \frac{1}{n(n+1)+1+{\overset{\infty }{\underset{k=1}{\Large\vcenter{\hbox{K}}}}~\dfrac{-k^4}{k^2+(k+1)^2+n(n+1)}}}$$

I guess the form of the continued fraction "tails" for $\eta(3)$ could be like:

$$ \eta(3)=\sum_{k=1}^{n}\frac{(-1)^{k-1}}{k^3}+(-1)^n\cdot \frac{1}{n^3+(n+1)^3+a+{\overset{\infty }{\underset{k=1}{\Large\vcenter{\hbox{K}}}}~\dfrac{\pm k^6}{p(k)+n^3+(n+1)^3}}}$$ where $p(k)$ is a polynomial of $k$, maybe of degree 3, and $p(0)=a$.

I worked on this format for several days, but haven't got anything good yet. The correct formula seems more complex than I expect. Does anyone kow this continued fraction "tails" for $\eta(3)$, or know how to derive such formula by referring the paper mentioned above?

Answer 1

Both authors of the paper you mention were students of mine, so I know the paper quite well! The method is called "Apery acceleration" and has been described in several places (if I am allowed some self-advertising, look at Section 7.5 of my joint book with K. Belabas "Numerical algorithms for number theory using Pari/GP", AMS Math surveys and monographs 254). To answer your specific question, we tried very hard in the 1980's to find a CF for the tails of $\eta(3)$ of the form you mention, with no success. I will soon post on the arXiv a much longer explanation of the method.