TITLE: Factorial and exponential dual identities
QUESTION [99 upvotes]: There are two identities that have a seemingly dual correspondence:
$$e^x = \sum_{n\ge0} {x^n\over n!}$$
and 
$$n! = \int_0^{\infty} {x^n\over e^x}\ dx.$$
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?

REPLY [8 votes]: Note that this duality is in fact a special case of Ramanujan's master theorem:
For some $x$ around the neighbourhood of $0$:
$$F(x) = \sum_{n=0}^\infty f(n) \frac{(-x)^n}{n!}$$
Then, 
$$\int_0^\infty x^{n} F(x)\,\text dx = n! f(-n-1)$$