Algebraic independence of certain infinite products involving the Fibonacci numbers

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.