My last post was on Panjer recursion, a recursive formula for the probability mass function of the compound distribution , where the are i.i.d. nonnegative-integer-valued random variables and is an independent counting process with -class distribution. In today’s brief post, I would like to use Panjer recursion to obtain a recursive formula for the moments of .

For ,

Applying the binomial expansion to and , we obtain

Substituting these identities in, we have

Moving the second term to the LHS and then dividing both sides by , we obtain

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