Irrationality and transcendence questions in the "poor man's ad\`ele ring"

TL;DR

This paper investigates arithmetic properties of the 'poor man's adèle ring' and proves a new transcendence result for sequences related to q-Fibonacci numbers, extending previous work that depended on the Generalized Riemann Hypothesis.

Contribution

It establishes the -transcendence of a sequence derived from q-Fibonacci numbers for integer q>1, generalizing prior results that required q to be square-free under GRH.

Findings

Proves -transcendence of the sequence for q>1.

Extends previous results beyond square-free q.

Relates transcendence to properties of the ade8le ring.

Abstract

We discuss arithmetic questions related to the "poor man's ad\`ele 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 $2\times2$-matrices $$ \bigg(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q^2 & 0 \end{matrix}\bigg) \cdots \bigg(\begin{matrix} 1 & 1 \\ q^{n-2} & 0 \end{matrix}\bigg) $$ and $q>1$ is an integer. This result was previously known for $q>1$ square free under the GRH.