See **NOTE** below.

This MO inquiry is over 3 yrs old now.

By the date **the question about the $\zeta(3)$ CF with $k=8/7$** was made (Feb, 2019), it can be answered in the negative nowadays, since it was '*(re)-discovered*' (and tagged as a *new* conjectured CF for Apéry's constant) by a team from Technion - Institute of Technology (Israel) using a highly specialized CAS named **The Ramanujan Machine**. See here. These findings for $\zeta(3)$ and several other constants were published in Arxiv (Table 5. pg. 16) in May, 2020 and also in Science Journal Nature (Feb, 2021). See Ref. below.

So, I think it deserves to be called Wolfgang's $\zeta(3)$ CF. It provides about 1.5 decimal digits per iteration.

Are there **other values of k** where such a polynomial exists?.

Answer is yes.
In addition to known $k=1,\,8/7,\,6$, ($k=6$ CF is equivalent to Apéry's recursion employed to prove $\zeta(3)$'s irrationality), $k=5/2$ is currently also known (June, 2019) and $k=12/7$ is conjectured (2020).

This youtube video shows at 26:10 $\zeta(3)$ Continued Fractions for $k=8/7$ (Wolfgang's) and $k=12/7$.

Must all those polynomials have a **zero at 1/2** for some deeper reason?

$k=5/2$ CF does not have a zero at 1/2, but $k=1,6,8/7,12/7$ do (all have companion numerator sequence polynomials $q(n)=-n^6$). There are some conjectures in this paper to look for candidates to prove (or improve) the irrationality (measure) of some constants based on the type of factors and roots that $q(n)$ must have.

**NOTE**.

As Wolfgang has pointed out in his answer, there was a "Séminaire de Théorie des Nombres" held in Bordeaux University on March 21th, 1980. In the Exposé N°23 by Christian Batut and Michael Olivier "Sur l'accéléracion de la convergence de certaines fractions continues" (in French), the $\zeta(3)$ CF with $k=8/7$ is found on page 23-20 3.2.4. In this presentation, Apéry's equivalent $\zeta(3)$ CF with $k=6$ is also found (23-19 3.2.2), together with some CFs for Catalan's and other constants. Other $\zeta(3)$ CFs like $k=5/2$ or $k=12/7$ are not shown.

To preserve the spirit of the original response, I will leave this answer at this point.

**Ref:** Raayoni, G., Gottlieb, S., Manor, Y. et al. Generating conjectures on fundamental constants with the Ramanujan Machine. Nature 590, 67–73 (2021). https://doi.org/10.1038/s41586-021-03229-4

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