Irrationality and transcendence questions in the 'poor man's adèle ring'

We discuss arithmetic questions related to the 'poor man's adèle ring' $A$ whose elements are encoded by sequences ${\left(t_p\right)}_p$ indexed by prime numbers, with each $t_p$ viewed as a residue in $Z/pZ$ . Our main theorem is about the $A$ -transcendence of the element ${\left(F_p\left(q\right)\right)}_p$ , where $F_n\left(q\right)$ (Schur's q-Fibonacci numbers) are the (1, 1)-entries of $2\times 2$ -matrices $\begin{array}{c}\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}1& 1\\ q& 0\end{array}\right)\left(\begin{array}{cc}1& 1\\ q^2& 0\end{array}\right)\cdots \left(\begin{array}{cc}1& 1\\ q^{n-2}& 0\end{array}\right)\end{array}$ and $q>1$ is an integer. This result was previously known for $q>1$ square free under the GRH.