Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers
Tom 168 / 2015
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.
