I often see here formula expressed in term of $z$. But what is $z$? While searching resources for generating pink noise (and with your help in the comments and answers of other questions), I came to such kind of formula:


  $$ H(z) = { .041 - .096z^{-1} + .051z^{-2} - .004z^{-3} \over{}  -2.495z^{-1} + 2.017z^{-2} - .522z^{-3}} $$ 
  
  (from https://dsp.stackexchange.com/a/376/46389)


What is $z$?

I know it should be obvious for anyone familiar with digital signal processing but for people coming from a different field, it is not obvious to know what this represents and how it relates to the signal amplitude or frequency or any other "tangible" parameter.



Edit: amusingly enough, the site suggested me the z-tranform tag for this question. I suspect I should consider that as a clue.
$z$ is interpreted as the time advance operator. $z^{-1}$ is the time delay operator.

So for a difference equation like
$$ y[n]=a x[n]+b x[n-1] $$
in the $z$ domain
$$ Y(z)=(a+ b z^{-1}) X(z)$$