Algebraic independence of certain infinite products involving the Fibonacci numbers

TL;DR

This paper expresses certain infinite products involving Fibonacci numbers in terms of Jacobi theta functions and proves their algebraic independence over the rationals.

Contribution

It provides explicit formulas for these infinite products using theta functions and establishes their algebraic independence, a novel connection between Fibonacci products and transcendence theory.

Findings

Explicit formulas for the infinite products in terms of theta functions

Proof of algebraic independence over  of the involved numbers

Application of Bertrand's theorem to Fibonacci-related constants

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.