Algebraic independence of infinite products generated by Fibonacci and Lucas numbers (Q2445027)

Let \(\alpha\) and \(\beta\) be algebraic numbers such that \(|\alpha| >1\) and \(\alpha\beta =-1\). Denote \[ U_n=\frac{\alpha^n-\beta^n}{\alpha -\beta}\quad \text{ and}\quad V_n=\alpha^n-\beta^n \] for every \(n\in \{0,1,\dots\} \). Let \(d_1\) and \(d_2\) be integers such that \(d_1, d_2\geq 2\) and let \(\gamma_1\) and \(\gamma_2\) be non-zero algebraic numbers with \((d_2,\gamma_2)\not= (2,-1), (2,2)\). Then the numbers \[ \prod^\infty_{\substack{ k=1\\ U_{d_1^k}\not= -\gamma_1 }} {\biggl(1+\frac {\gamma_1}{U_{d_1^k}}\biggr)}, \quad \text{and} \quad \prod^\infty_{\substack{ k=1\\ V_{d_2^k}\not= -\gamma_2 }} \biggl(1+\frac {\gamma_2}{V_{d_2^k}}\biggr), \] are algebraic independent over \(\mathbb Q\). The proof is based on the Mahler method.