Congruences of the $q$-Fibonacci sequence related with its transcendence

TL;DR

This paper establishes congruences connecting the $q$-Fibonacci sequence with the classical Fibonacci sequence and demonstrates the transcendence of certain elements in a specialized algebra under the generalized Riemann hypothesis.

Contribution

It introduces new congruences involving the $q$-Fibonacci sequence and proves the transcendence of specific algebraic elements assuming the generalized Riemann hypothesis.

Findings

Established congruences linking $q$-Fibonacci and Fibonacci sequences

Proved transcendence of certain elements in algebra $ extbf{A}$

Results depend on the generalized Riemann hypothesis

Abstract

By using Andrews's explicit formulae of the $q$-Fibonacci sequence introduced by Schur, we prove certain congruences of the $q$-Fibonacci sequence which relate the sequence with the original Fibonacci sequence. As a corollary, we show that it yields a transcendental element in the $\mathbb{Q}$-algebra $\mathscr{A}$ of integers modulo infinitely large primes under the generalized Riemann hypothesis.