It is somewhat common to write $f(x_0,x_1;\ z_0,z_1)$ to indicate that $f$ is a function of $x_0,\ x_1,\ z_0,\&\ z_0$, but that the $z$ arguments should be thought of as fixed by the context, while variation with respect to the $x$ arguments are core to the discussion.
For example when discussion the partial derivatives of $f$ with respect to the $x$'s for fixed $z$'s or the probability as a function of the $x$'s given the $z$'s.
In theorem proving this often feels like "a distinction without a difference", but in programing it may indicate the expectation of some sort of parameter specific optimization. The derived function $f_{z_0,z_1}(x_0,x_1)$ may be more efficiently implemented than the general function $f(x_0,x_1,z_0,z_1)$. Some systems have tried to automate the generation of an optimized derived function given parameter values, but they never really caught on.
I am writing and it would be extremely convenient to have a name for this. The passage is starting to feel repetitive and hard to read.
I don't want to create my own name, because I distinctly remember reading some reasonably authoritative source that referred to this as the "such-and-such" convention. So creating my own term would likely be a faux pas.
But I can't remember the name or re-find the passage.