Binomials whose dilations generate $H^2(\mathbb D)$

This note is about the completeness of the function families
$$ \{z^n(\lambda-z^n)^N\colon n=1,2,\dots\} $$
in the Hardy space $H^2_0(\mathbb D)$, and some related questions. It is shown that for $|\lambda|>R(N)$ the family is complete in $H^2_0(\mathbb D)$ (and often is a Riesz basis of $H^2_0$), whereas for $|\lambda|<r(N)$ it is not, where both radii $r(N)\leq R(N)$ tends to infinity and behave more or less as $N$ (as $N\to\infty$). Several results are also obtained for more general binomials $\{z^n(1-\frac1\lambda z^n)^\nu\colon n=1,2,\dots\}$ where $|\lambda|\geq1$ and $\nu\in\mathbb C$.