I'm stuck in the following problem and I need your help to solve it.
Given a number $\alpha$, $0 < \alpha < 1$. $A_j(x)$ is a sequence of polynomials of $x^{-1}$ such that:
$A_0(x) = 1; \\ A_{j+1}(x) = A_{j}(x) + \alpha^{j+1}x^{-j-1}A_j(x^{-1})\\ (A_1(x) = 1 + \alpha x^{-1}; A_2(x) = 1 + (\alpha + \alpha^3)x^{-1} + \alpha^2x^{-2},...)$
Prove that all roots of the equation $A_j(x) = 0, j \geq 1,$ lie on a circle of radius $\alpha$.
Could anyone give me some hints?
Thank you very much.