# Factorial and exponential dual identities

There are two identities that have a seemingly dual correspondence:

and

Is there anything to this comparison? (I vaguely remember a generating function/integration correspondence)

Are there similar sum/integration pairs for other well-known (or not-so-well-known) functions?

which is an application of the first identity. (This new identity is easy to prove, since the integrand is just $e^{(t-1)x}$ so it has antiderivative $\frac{1}{t-1} e^{(t-1)x}$ and the identity follows from here.)