Equations Involving Factorial

Question

Given polynomials $f$ and $g$, are there any known methods to solve Diophantine equations of the form $f(x)=g(x!)$ where $x!$ is the factorial of $x$? I have not been able to locate any resources regarding this. If such methods exist, what restrictions must be imposed on $f$ and $g$. I would be interested in learning about any results pertaining to solving these types of equations, or to how many solutions such equations might have. Any links to journal articles would also be greatly appreciated (JSTOR or open access would be preferred, since those are the only journals I currently can access).