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

Area of Science:

• Number Theory
• Algebraic Number Theory
• Arithmetic Geometry

Background:

• The study of arithmetic properties of sequences indexed by prime numbers.
• Introduction of the 'poor man's adèle ring' (A) for encoding such sequences.
• Previous results on A-transcendence for specific cases of q under the Generalized Riemann Hypothesis (GRH).

Purpose of the Study:

• To investigate the A-transcendence of a specific sequence related to q-Fibonacci numbers.
• To extend existing results on A-transcendence to a broader range of parameters.
• To contribute to the understanding of arithmetic properties within the 'poor man's adèle ring'.

Main Methods:

• Utilizing concepts from algebraic number theory and the theory of transcendental numbers.
• Defining Schur's q-Fibonacci numbers via matrix products.
• Analyzing the structure of elements within the 'poor man's adèle ring'.

Main Results:

• The main theorem establishes the A-transcendence of the sequence (F_p(q))_p, where F_p(q) represents Schur's q-Fibonacci numbers.
• This result generalizes previous findings, particularly for integer values of q > 1.
• The proof relies on the arithmetic properties of the defined ring and the nature of the q-Fibonacci sequence.

Conclusions:

• The A-transcendence of Schur's q-Fibonacci numbers is confirmed for integer q > 1.
• This work deepens the understanding of arithmetic in the 'poor man's adèle ring'.
• The findings have implications for the study of transcendental number theory in the context of number-theoretic sequences.