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The following series is related to Example X of the paper 2312.14051v4:

$$\sum_{k=0}^{\infty}\frac{ (-1)^k (5532k^4 + 5464k^3 + 2173k^2 + 416k + 32)\binom{2k}{k}^9}{ (10k + 1)(10k + 3)\binom{10k}{5k} \binom{5k}{k} \binom{4k}{k} \binom{3k}{k}}=\frac{80}{\pi^2} \tag{1} $$

In principle, it is very likely to be proven using the WZ method.

I have the following two questions:

  1. Is it known?

  2. Can it also be proven using the method answered by Jesús Guillera in here.

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1 Answer 1

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It is a translation of the formula X that you are citing. Just replace G(n,k) with G(n+1/2, k) and obtain the new F(n, k), and you will have a proof of it.

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  • $\begingroup$ Is it the same as the 21st formula on page 10 of the paper 2101.12592v1 arxiv.org/abs/2101.12592v1 ? $\endgroup$ Commented Feb 2 at 14:44
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    $\begingroup$ No, it is not the same. The formula in my paper joint with Cohen is a "divergent" (convergent by analytic continuation) for 1/π⁴. It is the dual of the formula for ζ(5) and not a translation. $\endgroup$ Commented Feb 2 at 15:02

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