# Transcendence of the $p$-adic number $\sum_{n\ge0}a^{2^n}$

Let $p$ a prime number and $a\in\overline{\mathbb Q}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$.

Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb Q$?

In the complex case, it is known that it is true (see Duverney Théorie des nombres or Nishioka Mahler functions and transcendence) But I can not find any proof or reference for the $p$-adic case.

Thanks in advance for any hints or references for this problem.

• OOPS. You are right. I meant $a\in\mathbb Q$. I edit my post. – joaopa Dec 29 '16 at 15:15
• I wonder if the first two theorems in chapter 67 of Schikhof's Ultrametric Calculus would be helpful to you. They are related to $p$-adic Liouville numbers over $\mathbb{Z}_p$. – gobucksmath Jan 17 '17 at 14:21