Non integer, non-centered Gaussian moments

Question

I have read the following question : Non-centered Gaussian moments where it is stated that :

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

I also read : Moments and Absolute Moments of the Normal Distribution that states the same thing.

Does this still hold if $p$ is not an integer ? I went through equations 27-32, of the paper, they do not seem to require $p$ to be an integer, but I am not very familiar with hyper-geometric functions.

Answer 1

In the paper, the variable you call $p$ is called $\nu$, and they state the results are valid for $\nu > -1$, with special cases given for integer $\nu$ ($\nu \in \mathbb{N}_0$ in the notation if the paper).

Given all this I think the result holds for non-integer $\nu$ ($p$), but I didn't go through the maths to be sure.

Hope this helps.