How to do this inverse Z-transform? Is this correct? It was explained to me here that I need to do an inverse z-transform on a z-based equation in order to get something I can use. That example was quite simple. But I'm not sure how to do the same process on a more complex equation.

I am starting with:

$H(s) = \frac{-ms}{ms+2R}$

Where m=mass and R=impedance. I can do a backward Euler substitution $s=\frac{1−z^{−1}}{T}$:

$H(z) = \frac{m (z^{-1} - 1)}{-m z^{-1} + m + 2 R T}$

How do I then get this in terms of $[n]$ and $[n-1]$? I understand $H(z) = H[n]$ and $H(z)*z^{-1} = H[n-1]$, but how do I mathematically accomplish this and get rid of the $z^{-1}$ terms here?

Here is my best try:

$-H(z) m z^{-1} + H(z) m + 2 R T H(z) = m z^{-1} - m$

$-H[n-1] m - H[n] m + 2 R T H[n] = m z^{-1} - m$

$H[n] (-m + 2 R T) = m z^{-1} - m + H[n-1] m$

$H[n] = \frac{m z^{-1} - m + H[n-1] m}{2 R T - m}$

And I'm stuck there with one remaining $z^{-1}$ term. Multiplying by $z^{-1}$ is supposed to be the same as delaying by a sample right? So if mass is constant, then can I just say for the remaining term $m z^{-1} = m$ and get:

$H[n] = \frac{H[n-1] m}{2 R T - m}$

Is this the right idea or if I screwed it up what was I supposed to do differently?

Thanks.
A rational transfer function

$$H(z)=\frac{Y(z)}{X(z)}=\frac{\sum_{n=0}^{N}b[n]z^{-n}}{1+\sum_{n=1}^{N}a[n]z^{-n}}\tag{1}$$

corresponds to the following difference equation:

$$y[n]=b[0]x[n]+b[1]x[n-1]+\ldots +b[N]x[n-N]-\\-a[1]y[n-1]-\ldots -a[N]y[n-N]\tag{2}$$

So the current output sample $y[n]$ can be computed from the current input sample $x[n]$ and $N$ past input and output samples. $N$ is the order of the difference equation $(2)$.

In your example you have

$$H(z)=\frac{Y(z)}{X(z)}=-c\frac{1-z^{-1}}{1-cz^{-1}},\qquad c=\frac{1}{1+\frac{2RT}{m}}\tag{3}$$

and, consequently,

$$y[n]=c\big(-x[n]+x[n-1]+y[n-1]\big)\tag{4}$$