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Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers

Hajime KanekoTakeshi KurosawaYohei TachiyaTaka-aki Tanaka — 2015

Acta Arithmetica

Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products k = 1 U d k - a i ( 1 + ( a i ) / ( U d k ) ) (i=1,...,m) or k = 1 V d k - a i ( 1 + ( a i ) ( V d k ) (i=1,...,m) to be algebraically dependent, where a i are non-zero integers and U n and V n are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers a 1 , . . . , a m to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically dependent....

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