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

Hajime Kaneko; Takeshi Kurosawa; Yohei Tachiya; Taka-aki Tanaka

Hajime Kaneko ; Takeshi Kurosawa ; Yohei Tachiya ; Taka-aki Tanaka

Acta Arithmetica, Tome 168 (2015), p. 161-186 / Harvested from The Polish Digital Mathematics Library

Résumé

Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products $\prod_{\binom{k = 1}{U_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) / (U_{d^{k}}))$ (i=1,...,m) or $\prod_{\binom{k = 1}{V_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) (V_{d^{k}})$ (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.

Publié le : 2015-01-01

Zbl 1331.11057

EUDML-ID : urn:eudml:doc:279621

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     author = {Hajime Kaneko and Takeshi Kurosawa and Yohei Tachiya and Taka-aki Tanaka},
     title = {Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers},
     journal = {Acta Arithmetica},
     volume = {168},
     year = {2015},
     pages = {161-186},
     zbl = {1331.11057},
     language = {en},
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Hajime Kaneko; Takeshi Kurosawa; Yohei Tachiya; Taka-aki Tanaka. Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers. Acta Arithmetica, Tome 168 (2015) pp. 161-186. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-2-5/