Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers

TL;DR

This paper explicitly evaluates two algebraic infinite products involving Fibonacci and Lucas numbers, building on Tachiya's 2007 conditions for their transcendence, and provides exact values for these products.

Contribution

It explicitly computes specific algebraic infinite products involving Fibonacci and Lucas numbers, expanding on Tachiya's earlier transcendence criteria.

Findings

Evaluated the infinite product involving Fibonacci numbers as 3/phi.

Evaluated the infinite product involving Lucas numbers as 3 - phi.

Confirmed the algebraic nature of these infinite products.

Abstract

In 2007, Tachiya gave necessary and sufficient conditions for the transcendence of certain infinite products involving Fibonacci numbers $F_k$ and Lucas numbers $L_k$. In the present note, we explicitly evaluate two classes of his algebraic examples. Special cases are$$\prod_{n=1}^{\infty}(1+\frac{1}{F_{2^n+1}})=\frac{3}{\phi}, \qquad \prod_{n=1}^{\infty}(1+\frac{1}{L_{2^n+1}})=3-\phi\, ,$$where $\phi=(1+\sqrt{5})/2$ is the golden ratio.