# Inverse $\mathcal Z$-transform of system with an 8th order pole

Can I find the inverse $\mathcal Z$-transform of this transfer function: $$H(z)=\frac{1}{1-\alpha z^{-8}}$$ in a way other than contour integration and finding the residues of the 8 poles? If so, how?

$$H(z) = \frac{1}{1-\alpha z^{-8}} = \sum_{n=0}^{\infty}(\alpha z^{-8})^n = \sum_{n=0}^{\infty}\alpha^n z^{-8n}$$ If $h_n = \mathcal Z^{-1}\{H(z)\}$, then $$h_n = \begin{cases} a^{n/8} \; , \; \text{n is multiple of 8} \\ 0 \; , \; \text{otherwise} \end{cases}$$