# Irrationality and transcendence questions in the ‘poor man’s adèle ring’

## TL;DR

This paper investigates mathematical properties of a special ring structure and proves a new result about the transcendence of a specific sequence of numbers.

## Contribution

The paper proves a new theorem about the A-transcendence of Schur’s q-Fibonacci numbers for integers q > 1.

## Key findings

- The element (F_p(q))_p is A-transcendental for integers q > 1.
- The result generalizes a previous finding that was only valid under the GRH and for square-free q.

## Abstract

We discuss arithmetic questions related to the ‘poor man’s adèle ring’ \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}$$\end{document}A whose elements are encoded by sequences \documentclass[12pt]{minimal}
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\begin{document}$$(t_p)_p$$\end{document}(tp)p indexed by prime numbers, with each \documentclass[12pt]{minimal}
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\begin{document}$$t_p$$\end{document}tp viewed as a residue in \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}/p\mathbb {Z}$$\end{document}Z/pZ. Our main theorem is about the \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}$$\end{document}A-transcendence of the element \documentclass[12pt]{minimal}
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\begin{document}$$(F_p(q))_p$$\end{document}(Fp(q))p, where \documentclass[12pt]{minimal}
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\begin{document}$$F_n(q)$$\end{document}Fn(q) (Schur’s q-Fibonacci numbers) are the (1, 1)-entries of \documentclass[12pt]{minimal}
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\begin{document}$$2\times 2$$\end{document}2×2-matrices \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ q & 0 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ q^2 & 0 \end{pmatrix} \cdots \begin{pmatrix} 1 & 1 \\ q^{n-2} & 0 \end{pmatrix} \end{aligned}$$\end{document}111011q011q20⋯11qn-20and \documentclass[12pt]{minimal}
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\begin{document}$$q>1$$\end{document}q>1 is an integer. This result was previously known for \documentclass[12pt]{minimal}
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\begin{document}$$q>1$$\end{document}q>1 square free under the GRH.