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

### Tom 168 / 2015

Acta Arithmetica 168 (2015), 161-186 MSC: Primary 11J85; Secondary 11J81, 11J91. DOI: 10.4064/aa168-2-5

#### Streszczenie

Let $d\geq 2$ be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products $$\prod_{\textstyle {k=1\atop U_{d^k}\neq-a_i}}^{\infty}\biggl( 1+\frac{a_i}{U_{d^k}}\bigg)\quad (i=1,\dots,m)\quad {\rm or} \!\quad\prod_{\textstyle{k=1\atop V_{d^k}\neq-a_i}}^{\infty}\biggl( 1+\frac{a_i}{V_{d^k}}\bigg)\quad (i=1,\dots,m)$$ 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 $a_1,\dots,a_m$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically dependent.

#### Autorzy

• Hajime KanekoInstitute of Mathematics
University of Tsukuba
1-1-1, Tennodai
Tsukuba, Ibaraki 350-0006, Japan
e-mail
• Takeshi KurosawaDepartment of Mathematical Information Science
Tokyo University of Science
1-3, Kagurazaka, Shinjuku-ku
Tokyo 162–8601, Japan
e-mail
• Yohei TachiyaGraduate School of Science and Technology
Hirosaki University
Hirosaki 036-8561, Japan
e-mail