syntax – 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?