# irrationality of Bessel function in $p$-adic

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.

• $J_0(1/k)$ is irrational. What does it have anything to do with the $p$-adics ? – reuns Feb 14 at 5:31
• @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 $\mathbb Q$. – Wojowu Feb 14 at 9:11