Application of the generating function

The kinetic scheme outlined in section 3.3 for anionic chain-growth polymerization resulted in the following mathematical representation for the evolution of monomer and polymeric species  

The last term can be simplified using the shifting theorem (equation (3.30)) to obtain 

Equation (3.41) is a generating function of Pj. The transform can be readily inverted by expanding the solution (3.41) in a Laurent s series in s and equating terms of order n in s with P,. The result is 

This is the Poisson distribution, the properties of which are discussed in most elementary statistics textbooks. It is the distribution arising in any situations when M objects (i.e. monomers) are divided among /g categories (i.e. active chains). The expected (mean) number per category is x and P gives the frequency distribution per category. 

The moment generating property can be particularly useful in the problem considered here. Although we could have obtained the moments of the distribution by formulating the moment equations and solving them, the moment generating property of generating functions offers ready access to the moments. Equation (3.41) can be restated as 

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