calculus and analysis - Derivative of generating function (Example from documentation)

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.

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