  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.
