I wonder if there is some reference for materials like the following.
Let $\frac{1}{\beta(z)}$ be well-defined for all z with $|z|\leq1$, where $\beta(z)=\sum_{k=0}^{\infty} \beta_kz^k$.
Then, we already know that $\frac{1}{\beta(z)}$ can be expressed as $\frac{1}{\beta(z)}=\alpha(z) = \sum_{k=0}^{\infty}\alpha_kz^k$, where $\sum_{k=0}^{\infty}|\alpha_k|<\infty$, which is frequently used for invertibility of an ARMA process in time-series.
But, I encountered in a paper the sentence that "further, if $\sum_{k=0}^{\infty}|k|^s|\beta_k|<\infty $ for some $s\geq1$, then we have that $\sum_{k=0}^{\infty}|k|^s|\alpha_k|< \infty$, which I have not yet experienced studying mathematics and time-series.