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arXiv · 2009.06250

Algebraic independence of certain infinite products involving the Fibonacci numbers

Daniel Duverney·Yohei Tachiya
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Abstract

Let $\{F_{n}\}_{n\geq0}$ be the sequence of the Fibonacci numbers. The aim of this paper is to give explicit formulae for the infinite products \[ \prod_{n=1}^{\infty}\left( 1+\frac{1}{F_{n}}\right) ,\qquad\prod_{n=3}^{\infty}\left( 1-\frac{1}{F_{n}}\right) \] in terms of the values of the Jacobi theta functions. From this we deduce the algebraic independence over $\mathbb{Q}$ 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. 2020-09-14. Algebraic independence of certain infinite products involving the Fibonacci numbers. https://arxiv.org/abs/2009.06250

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