Transcendence of certain infinite products (Q884529)

Let \(r\geq 2\) and \(L\geq 1\) be integers, \((E_{k})_{k\geq 0}\) and \((F_{k})_{k\geq 0}\) sequences of polynomials of degree \(\leq L\) with coefficients in a number field such that \(E_{k}(0)=F_{k}(0)=1\). The author considers the function \[ \Phi_{0}(x)= \prod_{k\geq 0} E_{k}(x^{r^{k}})/F_{k}(x^{r^{k}}) \] for \(| x| <1\). Let \(\alpha\) be an algebraic number such that \(0<| \alpha| <1\) and \(E_{k}(\alpha^{r^{k}})F_{k}(\alpha^{r^{k}})\not=0\) for all \(k\geq 0\). Under usual assumptions on the growth of the heights and denominators of the coefficients of the polynomials \(E_{k}\) and \(F_{k}\), he shows that \(\Phi_{0}(\alpha)\) is algebraic if and only if \(\Phi_{0}(x)\) is a rational function of \(x\). Several related results are also established. These results dealing with infinite products are analogs to those for infinite series obtained by [\textit{D.~Duverney} and \textit{K.~Nishioka} in [Acta Arith. 110, No. 4, 305--330 (2003; Zbl 1049.11078)].