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In general, the list of arguments of a function includes only variables, not parameters.

In some specific cases, a parameter could be incorporated into the function name, like $y$ in $$\log_y (x)$$

But is there any more or less standard notation for parameters?

What I've found so far is that in probability theory very often parameters are delimited from variables with semicolon, i.e. $f(x; y)$. However I have some objects which are not functions but they parametrically depends on some other objects and it is impossible to show such dependence with this notation.

Alternative approach (found in some physics books) is to enclose parameters in the arguments list in curly brackets, i.e. $f(x, \{y\})$. Then if some object which is not a function but depends on some parameter can be denoted as $O(\{y\})$. But this notation is confusing since $\{ \}$ are used quite frequently for sets.

Is there something else?

P.S. I don't want to use subscripts or superscripts because they are already used for other purposes.

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You can use, for example, the ";" separator or the "|"-separator: they are commonly used in statistics. For example people write

$$f(x;\mu,\sigma)=\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{(x-\mu)^2}{2\sigma^2}}, $$

where $x$ is the variable for the density $f$, while $\mu$ and $\sigma$ are the parameters. Similarly, you can introduce $f(x|\mu,\sigma)$.

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