I found the expression for the expected value of the falling factorial of a Poisson distribution ($\lambda^n$) from  http://en.wikipedia.org/wiki/Factorial_moment. Is there a similar expression for the expected values of the rising factorial and plain simple factorial? I am actually after the expected value of the logarithms of these quantities: $$\sum_{u=1}^{X}log(u)$$ and $$\sum_{u=X}^{X+c}log(u)$$ where $X \to Poisson(\lambda)$.
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