Algebraic independence of infinite products generated by Fibonacci numbers (Q533736)

The main result of this paper is the following theorem. Theorem: Let \(m\) and \(d\) be a positive integers such that \(d>2\). Assume that \(a_1, a_2,\dots , a_m\) are nonzero distinct integers. Suppose that \(\{R_n\}_{n=1}^\infty\) is a linear recurrence of the second order which fulfils some other conditions. Then the numbers \(\prod_{n=1}^\infty (1+\frac {a_i}{R_{d^n}})\), \(i=1,\dots ,m\) are algebraically independent. If \(R_n\neq -a_n\) for all \(n=1,\dots\) then this theorem also holds when \(\{R_n\}_{n=1}^\infty\) is Fibonacci or Lucas sequence.