Abstract
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 \(\mathbb {Q}\)-algebra \(\mathscr {A}\) of integers modulo arbitrarily large primes under the generalized Riemann hypothesis.
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Acknowledgements
T.A. would like to take this opportunity to thank the “Interdisciplinary Frontier Next-Generation Researcher Program of the Tokai Higher Education and Research System”. We are deeply grateful to Hidekazu Furusho; without his profound instruction and continuous encouragement, this paper would never be accomplished. We are grateful to Henrik Bachmann, Minoru Hirose, Toshiki Matsusaka, Leo Murata, Shin-ichiro Seki, Koji Tasaka and Shuji Yamamoto for answering our questions. We would like to thank Jun Ueki for giving us much advice on the structure of this paper.
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Anzawa, T., Funakura, H. Congruences for the q-Fibonacci sequence related to its transcendence. Ramanujan J 63, 1057–1072 (2024). https://doi.org/10.1007/s11139-023-00802-5
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DOI: https://doi.org/10.1007/s11139-023-00802-5
Keywords
- Finite algebraic number
- Finite transcendental number
- q-Fibonacci sequence
- Finite multiple zeta value
- Transcendental number