Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$

Question

For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial in finite fields. Are there analogues of binomial coefficients which can be represented nicely? Would it be reasonable to seek analogues which can be represented nicely of denominator of Bernoulli numbers $B_{2n}$ which are $\prod_{(p-1)|2n}p$?