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Let $J_0(z)=\sum_{n\ge 0}\frac{(-1)^n}{n!^2}\left(\frac z2\right)^{2n}$ be the Bessel function considered in $\mathbb C_p$. Let $\alpha\in\mathbb Q^*$ be in the convegence disk of $J_0$. Is $J_0(\alpha)$ irrational? That sounded a pretty natural question, but I find nothing about this in googling. Any answer will be welcome.

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  • $\begingroup$ $J_0(1/k)$ is irrational. What does it have anything to do with the $p$-adics ? $\endgroup$ – reuns Feb 14 at 5:31
  • $\begingroup$ @reuns You can consider the sum of the provided series in the $p$-adic numbers (as long as it converges). Irrationality of the value when considered in real numbers doesn't necessarily have any connection to whether the $p$-adic sum is in $\mathbb Q$. $\endgroup$ – Wojowu Feb 14 at 9:11

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