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

Hajime Kaneko¹ ; Takeshi Kurosawa² ; Yohei Tachiya³ ; Taka-aki Tanaka⁴

¹ Institute of Mathematics University of Tsukuba 1-1-1, Tennodai Tsukuba, Ibaraki 350-0006, Japan
² Department of Mathematical Information Science Tokyo University of Science 1-3, Kagurazaka, Shinjuku-ku Tokyo 162–8601, Japan
³ Graduate School of Science and Technology Hirosaki University Hirosaki 036-8561, Japan
⁴ Department of Mathematics Keio University 3-14-1, Hiyoshi, Kohoku-ku Yokohama 223-8522, Japan

Acta Arithmetica, Tome 168 (2015) no. 2, pp. 161-186

Résumé

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.

Détail

DOI : 10.4064/aa168-2-5

Keywords: geq integer second third fourth authors gave necessary sufficient conditions infinite products prod textstyle atop neq a infty biggl frac bigg quad dots quad quad prod textstyle atop neq a infty biggl frac bigg quad dots algebraically dependent where non zero integers and generalized fibonacci numbers lucas numbers respectively purpose paper relax condition non zero integers dots non zero real algebraic numbers which gives cases where infinite products above algebraically dependent