Is there an identity for simplifying expressions of the form $$ (n+m)!(n-m)! - n!^2 $$ or anything resembling this?
It would be particularly nice if the resulting expression was a linear combination of factorials, whose coefficients are given in terms of $n$ and $m$. (This problem originates in simplifying some differences of hypergeometric functions.) If that doesn't exist, then something using powers of $m$ and $n$ might work.
I imagine that something of the latter kind could probably be done with Stirling numbers, but I wanted to see if anyone is aware of an existing derivation.