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.

  • 2
    $\begingroup$ Same in 9.0.1 Windows. ZTransform behavior similarly changed it appears. $\endgroup$ – ciao Apr 12 '14 at 22:28
  • 1
    $\begingroup$ The issue persists in the Raspberry Edition of Mma v10. $\endgroup$ – Eckhard Apr 19 '14 at 19:20
  • $\begingroup$ This is a bug either of the function or of the documentation. I have tagged it accordingly. Whether the change in behavior was intentional or not I cannot say. $\endgroup$ – Mr.Wizard Aug 18 '14 at 23:22

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