Expectation values for integer powers of a Poisson-distributed random number

What is it about?

It is shown that expectation values of Poisson-distributed random numbers exist not only for the well known positive integer powers but also for negative integer powers. A recursion formula for the calculation of expectation values of powers differing by one is given. This recursion formula helps to find an analytical representation for both positive and negative integer powers in terms of the hypergeometric function.

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