Factorial and exponential dual identities

There are two identities that have a seemingly dual correspondence:

ex=n0xnn!

and

n!=0xnex 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?

Answer

There is a close relationship between the two identities, but I don’t know if the exact formal similarity is anything other than a neat coincidence along the lines of the Sophomore’s dream (although I could of course be wrong about this). First note that the second identity can be written as

1=0exxnn!dx

and therefore it is equivalent to the identity

11t=n=0tn0exxnn!dx=0exetxdx.

which is an application of the first identity. (This new identity is easy to prove, since the integrand is just e(t1)x so it has antiderivative 1t1e(t1)x and the identity follows from here.)

I know of interesting explanations of the two identities separately which are somewhat related, but not another direct connection like the one above: for the first see this math.SE question and for the second see this math.SE question.

Attribution
Source : Link , Question Author : Mitch , Answer Author : Community

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