Hokkaido Mathematical Journal

LUCA Florian, TACHIYA Yohei,

Algebraic independence of infinite products generated by Fibonacci and Lucas numbers.

Hokkaido Mathematical Journal, 43 (2014) pp.1-20

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Abstract

The aim of this paper is to give an algebraic independence result for the two infinite products involving the Lucas sequences of the first and second kind. As a consequence, we derive that the two infinite products Πk=1∞(1+1/F2k) and Πk=1∞(1+1/L2k) are algebraically independent over ℚ, where {Fn}n≥0 and {Ln}n≥0 are the Fibonacci sequence and its Lucas companion, respectively.

MSC(Primary)11J85
MSC(Secondary)11B39
Uncontrolled KeywordsInfinite products, algebraic independence, Mahler-type functional equation, Fibonacci numbers