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      Algebraic independence of certain infinite products involving the Fibonacci numbers

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          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.

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          Author and article information

          Journal
          14 September 2020
          Article
          2009.06250

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

          Custom metadata
          11J85, 11B39, 11F27
          4 pages
          math.NT

          Number theory

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