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

Volume 168 / 2015

Hajime Kaneko, Takeshi Kurosawa, Yohei Tachiya, Taka-aki Tanaka Acta Arithmetica 168 (2015), 161-186 MSC: Primary 11J85; Secondary 11J81, 11J91. DOI: 10.4064/aa168-2-5


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.


  • Hajime KanekoInstitute of Mathematics
    University of Tsukuba
    1-1-1, Tennodai
    Tsukuba, Ibaraki 350-0006, Japan
  • Takeshi KurosawaDepartment of Mathematical Information Science
    Tokyo University of Science
    1-3, Kagurazaka, Shinjuku-ku
    Tokyo 162–8601, Japan
  • Yohei TachiyaGraduate School of Science and Technology
    Hirosaki University
    Hirosaki 036-8561, Japan
  • Taka-aki TanakaDepartment of Mathematics
    Keio University
    3-14-1, Hiyoshi, Kohoku-ku
    Yokohama 223-8522, Japan

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