# Probability Generating Function(s)?

I'm very new to Mathematica and I have started to use it since I have been tackling some Physics related problems(I'm an engineer by training).

Specifically, I'm dealing with Master Equations and one of the tools used to try to solve Master equations under certain conditions are Probability generating functions. i.e.

$$T(z) = \sum_{a=0}^{\infty} z^a P(a)$$

where $$P(a)$$ is a probability mass function with discrete support, e.g. Poisson distribution

• Is there a way to compute such transformation?
• Should I use the Z-transform which has a similar definition?
• Can somebody show me how the problem would be approached as I'm very new to the syntax?
• It would be helpful if you provided some context and an example. I for one have literally no idea what Master equations and probability generating functions are. I would recommend showing an example in detail, and perhaps showing what you’ve tried so far in MMA. Commented Nov 26, 2020 at 15:54
• Hi Marco, I will soon post both an example as well as the solution.
– Jpk
Commented Nov 26, 2020 at 15:59

I found the answer in some documentation. The function to be used is called GeneratingFunction.

Consider for example the following function - a term of a Master equation - :

$$aP(a)$$.

We can transform such function by making use of probability generating functions with the following syntax:

GeneratingFunction[a*P[a], a, z]


z*D[GeneratingFunction[P[a], a, z], z]