In the appendix of the book http://www.math.upenn.edu/~wilf/DownldGF.html, page 194, the author mentioned that in Mma 2.0 (very old. :P), there is the function which can get the generating functions directly from recurrence relationship, like the example given therein:
GeneratingFunction[{f[n+2]==f[n+1]+f[n], f[0]==0, f[1]==1},f[n],n,x]
But the same function does not seem to do the kind of job anymore in mma 7.0/8.0. Does anyone know how to get the equivalent function? many thanks.
The GeneratingFunction scope has changed. Here you may find the obsolete legacy documentation (by the middle of the document).
Now you can do the same, but in two steps. First solve the recurrence relation with RSolve and the find the Generating Function. Like this:
GeneratingFunction[
RSolve[{f[n + 2] == f[n + 1] + f[n], f[0] == 0, f[1] == 1}, f[n], n],
n, x]
Out
{{GeneratingFunction[f[n], n, x] -> -(x/(-1 + x + x^2))}}
RSolve[] since that will actually solve the recurrence, when all you really need is the knowledge of the difference equation. The documentation has the identical example: GeneratingFunction[DifferenceRoot[Function[{y, m}, {y[m + 2] == y[m + 1] + y[m], y[0] == 0, y[1] == 1}]][n], n, z]
