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May 2021 Algebraic independence of certain infinite products involving the Fibonacci numbers
Daniel Duverney, Yohei Tachiya
Proc. Japan Acad. Ser. A Math. Sci. 97(5): 29-31 (May 2021). DOI: 10.3792/pjaa.97.006

Abstract

Let $\{F_{n}\}_{n\geq0}$ be the Fibonacci sequence. The aim of this paper is to give explicit formulae for the infinite products $$\begin{equation*} \prod_{n=1}^{\infty}\left( 1+\frac{1}{F_{n}}\right) ,\quad\prod_{n=3}^{\infty}\left( 1-\frac{1}{F_{n}}\right) \end{equation*}$$ in terms of the values of the Jacobi theta functions. From this we deduce the algebraic independence over $\mathbf{Q}$ of the above numbers by applying Bertrand’s theorem on the algebraic independence of the values of the Jacobi theta functions.

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Daniel Duverney. Yohei Tachiya. "Algebraic independence of certain infinite products involving the Fibonacci numbers." Proc. Japan Acad. Ser. A Math. Sci. 97 (5) 29 - 31, May 2021. https://doi.org/10.3792/pjaa.97.006

Information

Published: May 2021
First available in Project Euclid: 27 April 2021

Digital Object Identifier: 10.3792/pjaa.97.006

Subjects:
Primary: 11B39, 11F27, 11J85

Rights: Copyright © 2021 The Japan Academy

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Vol.97 • No. 5 • May 2021
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