Algebraic independence of certain infinite products involving the Fibonacci numbers
| Autor: | Duverney, Daniel, Tachiya, Yohei |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 4 pages |
| Databáze: | arXiv |
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