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Consider the series $$f(z)=\sum_{n\ge1}\dfrac{z^n}{n^2\binom{2n}{n}}$$ which converges for $|z|\le4$.

One has $f(0)=0$, $f(1)=\pi^2/18$, $f(2)=\pi^2/8$, $f(3)=2\pi^2/9$, and $f(4)=\pi^2/2$. Naive question: are there any other reasonably "explicit" evaluations (for instance for $z=-1$, $-2$, $1/2$...) ? Note that the series for $z=1$ is related to Ap'ery's proof of the irrationality of $\zeta(2)$.

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I interpret the request of the OP for an "explicit" evaluation of the series as a request for a "closed-form" expression, which exists (it seems to go back to Euler, here are several proofs): $$f(z)=\sum_{n\ge1}\dfrac{z^n}{n^2\binom{2n}{n}}=2 \arcsin^2\,(\tfrac{1}{2}\sqrt{z}),\;\;|z|\leq 4.$$

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    $\begingroup$ The identity is also stated in the Wikipedia page on inverse trigonometric functions ( en.wikipedia.org/wiki/… ). A reference is given to the book "Experimentation in Mathematics: Computational Paths to Discovery" (1 ed.) by Borwein, Bailey and Gingersohn. A proof via hypergeometric functions is sketched on page 51 of that book. $\endgroup$
    – Ofir Gorodetsky
    Commented Mar 12, 2023 at 20:24
  • $\begingroup$ Thanks, sorry for my naivete, I probably knew this formula. Apologies. $\endgroup$
    – Henri Cohen
    Commented Mar 12, 2023 at 20:47

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