Can a probability generating function have a constant term?

[Poirot1]

My question is : can a pgf have a constant term?

The reason I ask is that I was asked to show the (time) derivative of a pgf was equal to some multiple of the pgf and hence show the pgf was as given. So naturally , I differentiated the given answer and showed it satisfied the equation. But surely I could stick a constant on the end and it would differentiate to the same thing.

 

[CaptainBlack]

My question is : can a pgf have a constant term?

The reason I ask is that I was asked to show the (time) derivative of a pgf was equal to some multiple of the pgf and hence show the pgf was as given. So naturally , I differentiated the given answer and showed it satisfied the equation. But surely I could stick a constant on the end and it would differentiate to the same thing.

The simple answer is yes.

The more complicated answer is: please post the actual question or more context.

Adding something onto a pgf will result in it no longer being a pgf, since the sum of the coefficients should be 1.

CB