Algebraic independence of certain infinite products involving the Fibonacci numbers
نویسندگان
چکیده
Let $\{F_{n}\}_{n\geq0}$ be the Fibonacci sequence. The aim of this paper is to give explicit formulae for 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 values Jacobi theta functions. From we deduce algebraic independence over $\mathbf{Q}$ above numbers by applying Bertrand’s theorem on
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ژورنال
عنوان ژورنال: Proceedings of the Japan Academy. Series A, Mathematical sciences
سال: 2021
ISSN: ['0386-2194']
DOI: https://doi.org/10.3792/pjaa.97.006