I would like to know how to apply a low pass filter (Butterworth) to a digital signal. So, I have some values of the signal, let's say $S(t)$. Those values are equally spaced in time. I have read the Wikipedia's article, but I do not understand how to apply the transfer function $H$. I assume that I have to use a Z-transformation and apply it on $S$ but I am not certain to know how to do that. By the way, the part Digital implementation of this article is not really detailed.
Can someone help me to understand those things ? I would really appreciate any answers :)
P.S. I know there are functions in Python (e.g. scipy) that handle that issue, but I would like to know the mathematics behind it.