One of the main ingredients in Apéry's proof of the irrationality of $\zeta(3)$ is the existence of the fast-converging series:

$$ {\displaystyle {\begin{aligned}\zeta (3)&={\frac {5}{2}}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{{\binom {2k}{k}}k^{3}}}\end{aligned}}}. $$

Despite numerous attempts, no similar expressions were found for other values of the Riemann $\zeta$-function at positive odd integers.

For Catalan's constant, however, we do have such an expression, namely:

$${\displaystyle G={\frac {\pi }{8}}\log \left(2+{\sqrt {3}}\right)+{\frac {3}{8}}\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}{\binom {2n}{n}}}}.}$$

Why is this not sufficient for applying an Apéry-like method for proving its irrationality?

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    $\begingroup$ Having such an expression is by no means the same as irrationality. It is not clear how to adapt Apery's proof. $\endgroup$ – Dietrich Burde Oct 8 '18 at 12:42

Maybe the central binomial series I've discovered in arXiv paper https://arxiv.org/abs/1207.3139 should be useful for you. There in that paper I've shown that the convergence rate of my series is better than that you mentioned above. Regards, Fabio


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