Rules for pgf

This type of coupling occurs quite often in probability problems and is called a convolution of two distributions (mostly abbreviated by a star product). Passing to generating functions, the rather complicated sum of products of probabilities is transformed into a simple product of the corresponding pgf-s. 

Actually, however, the probability of finding exactly i-units in the first generation is Nj(l), and the total distribution is therefore 

Note Nj (1) is the probability that the root has i- offcpring , and Nkj (2) is the probability for j- offspring of a unit in the first generation. 

Now multiplying Eq. (C.74) on both sides by s1 and summing over all i, we obtain after application of the convolution theorem 

Proceeding stepwise to higher generations, one finally obtains for the pgf of the n-th generation 

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