# Puzzling result for CharacterisicFunction and InverseFourierSequenceTransform

Observe:

cf = CharacteristicFunction[PascalDistribution[1, p], t];
rPDF = InverseFourierSequenceTransform[cf, t, n, FourierParameters -> {1, -1}];

FullSimplify[PDF[PascalDistribution[1, p], n] == rPDF, n >= 1]
FullSimplify[PDF[PascalDistribution[1, p], n] == FourierCoefficient[cf, t, n], n >= 1]


True

True

As expected, the IFST of the CF is equivalent to the PDF, and also equivalent to the FC of the CF.

Now,

cf = CharacteristicFunction[GeometricDistribution[p], t];
rPDF = InverseFourierSequenceTransform[cf, t, n, FourierParameters -> {1, -1}];

FullSimplify[PDF[GeometricDistribution[p], n] == rPDF, n >= 1]

FullSimplify[PDF[GeometricDistribution[p], n] == FourierCoefficient[cf, t, n],  n >= 1]


results in: and

True

The returned expression under simplification using various assumptions returns nonsense, and itself is nonsense (e.g. it contains an assumption implying probability p >1 or p<0). I also manipulated the FourierParameters to no avail.

The example with GeometricDistribution returns correct results if the parameter p is explicitly specified as some valid probability.

I re-read the appropriate documentation entries to check if my recollection was faulty (I'm quite sure I used this functionality for GeometricDistribution before), and strangely the very example is the last entry for the Properties and Relations of CharacteristicFunction, where the correct results are shown, but if I re-evaluate the documentation example, I get the same nonsense.

Has something changed in InverseFourierSequenceTransform in recent versions (the above is on 9.1, Windows)?

## 1 Answer

It does appear something might have been tinkered with, or my recollection / documentation is in error.

InverseFourierSequenceTransform[cf, t, -n, FourierParameters -> {1, 1},
Assumptions -> 0 < p < 1] // PiecewiseExpand


properly recovers the PDF (PMF) for the GeometricDistribution case.