Bug introduced in 9.0 or earlier and persisting through 11.0.1 or later
In the documentation for
GeneratingFunction, the following example is given under Examples -> Properties & Relations -> Derivative:
GeneratingFunction[n f[n], n, z] (*z (GeneratingFunction^(0,0,1))[f[n],n,z]*)
When I evaluate the input, however, I obtain something different, namely:
GeneratingFunction[n f[n], n, z] (*z GeneratingFunction[(1 + n) f[1 + n], n, z]*)
For me the first output would be preferable; indeed I would like Mathematica to express any expression of the form
GeneratingFunction[p[n] f[n], n, z] (*p[n] an explicit polynomial in n*)
in terms of derivatives of the generating function of the sequence
f[n]. I think this is the intended behaviour of
GeneratingFunction and certainly what I remember from v8.
Question: Is there a way to achieve this behaviour?
I am currently using version 9.0 on a 64bit Linux computer.