How to find system function, H(z) in the z-domain, given zero-pole plot of the system? I'm given a zero-pole plot (Real/Imaginary Plane) of a system in which poles occur at $\frac{-9}{10}$, $\frac{9}{10}$, and zeros occur at $\frac{10}{9}j$, $\frac{-10}{9}j$. An additional zero occurs at negative infinity. 

I'm also given that $H_{F}(f)=1$ for $f=0$, where $H_{F}(f)$ refers to $H(z)$ for when $z=e^{-j 2\pi  f}$. I'm tasked to find $H(z)$.

I know I can express $H(z)$ as $$H(z)=G_0z^{-1}\frac{(z-z_0)(z-z_1)\ldots}{(z-p_0)(z-p_1)\ldots} $$ for zeros $$z_0,z_1,\ldots$$ and poles $$p_0, p_1,\ldots$$ 

I can find the constant $G_0$ by using the fact  $H_{F}(f)=1$ for $f=0$. 

I'm unsure how to express the fact that the system has a zero at negative infinity.
The $z^{-1}$ term in the $H(z)$ expression you provided is the expression of the zero at infinity:

$$\lim_{z\to\infty} \frac{1}{z} = 0$$

And in your specific case:

$$\lim_{z\to\infty} H(z) = G_{0}\frac{1}{z}\frac{(z-\frac{10}{9}j)(z+\frac{10}{9}j)}{(z-\frac{9}{10})(z+\frac{9}{10})} = 0$$

Generally, if the order of the denominator exceeds the order of the numerator, then you have 1 or more zeros at $\infty$