- Record: found
- Abstract: found
- Article: found

Preprint

Daniel Duverney , Yohei Tachiya

14 September 2020

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.

http://arxiv.org/licenses/nonexclusive-distrib/1.0/