Congruences for the q-Fibonacci sequence related to its transcendence

By using Andrews’ 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 \(\mathbb {Q}\) $\mathbb{Q}$ -algebra \(\mathscr {A}\) $\mathcal{A}$ of integers modulo arbitrarily large primes under the generalized Riemann hypothesis.