This was a surprising pair of identities I stumbled on in college.

and

I don’t remember the source of the question, but I suspect it was someone with knowledge of this relation.

I’m very curious whether this is part of a larger class of mathematical expressions involving roots that can be simplified.

Here’s a proof:

Solving for m and n:

Interestingly, this can be used in a telescoping identity when a=k, b=1: