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Algebraic independence of infinite products generated by Fibonacci and Lucas numbers. (English) Zbl 1291.11103
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.

##### MSC:
 11J85 Algebraic independence; Gel’fond’s method 11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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