Algebraic independence of certain infinite products involving the\n Fibonacci numbers

2020/09/14 by Daniel Duverney, Duverney, Daniel, Yohei Tachiya +1
Mathematics · #11B39 #11F27 #11J85 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2009.06250

openalex publication_date 2020/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F_{n n\≥0} be the sequence of the Fibonacci numbers. The aim of\nthis paper is to give explicit formulae for the infinite products n
prod_n=1
^infty
left( 1+
frac1F_n
right)\n,
qquad
prod_n=3
^infty
left( 1-
frac1F_n
right) in terms of the\nvalues of the Jacobi theta functions. From this we deduce the algebraic\nindependence over \ℚ of the above numbers by applying Bertrand's\ntheorem on the algebraic independence of the values of the Jacobi theta\nfunctions.\n

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