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# Algebraic independence of certain infinite products involving the Fibonacci numbers

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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.

### Author and article information

###### Journal
14 September 2020
###### Article
2009.06250

11J85, 11B39, 11F27
4 pages
math.NT

Number theory