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I would like to find a (closed nice) expression for the non-centered Gaussian moments with mean $\mu$ and variance $\sigma$. In found something in wikipedia:

$$E|X|^p = \sigma^p 2^{p/2} \frac{\Gamma \left(\frac{p+1}{2}\right)}{\sqrt{\pi}} {}_1 F_1 \left(-\frac{1}{2}p, \frac{1}{2}, -\frac{1}{2}(\mu/\sigma)^2\right)$$

but I don't understand this "confluent hypergeometric function" ${}_1F_1$ because it doesn't seem to be well-defined for negative integers according to this article in wikipedia.. (it seems to be only valid for positive integers)

Does anyone know about this? Thank you very much for the help!

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Suppose $Z\sim N(0,1)$ and $X=\mu+\sigma Z$, so that $X\sim N(\mu,\sigma^2)$.

$$ \mathbb E(X^n) = \mathbb E((\mu+\sigma Z)^n) = \sum_{k=0}^n \binom n k \mu^{n-k} \sigma^k \mathbb E(Z^k). $$

If you know the moments of $Z$, plug them in there.

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That could be one possibility yes, thanks! But I still wonder what they mean by ${}_1 F_1(a,b,c)$, it is strage.. i don't find anything related.. – Dan Mar 17 '13 at 18:52

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