given the one-zero digital filter,
$$y[n] = x[n]\cdot \frac{1}{2} + x[n-1] \cdot \frac{1}{2}$$
Which has the transfer function:
$$H(z) = (1+z^{-1})\cdot \frac{1}{2}$$
and taking $M$ of these filters in series as a cascade, giving a filter $G$ with
$$G(z) = H(z)^{M}$$ Is there a known closed form of G's impulse response?
I have calculated one and am not sure if it's documented somewhere. I know we can take the inverse fourier transform of $G(z)$, but we arrive at an integral/sum. I wonder about a closed form.
Thank you very much.