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