I am trying to find a closed solution to find the expected value of $X^p$ with $p>0$, where $x$ is drawn from a truncated normal distribution cut on both sides, such that it is non-zero only in a range from $a$ to $b$. I know that the plain moments of the normal distribution involving hypergeometric confluent functions are defined not only for integers, but also for $p>-1$, see here:
https://en.wikipedia.org/wiki/Normal_distribution#Moments
Can I exploit this to solve this also for a truncated normal distribution? To my understanding, the truncated normal distribution has the same PDF as the normal ($\rho_N$), except that it is zero outside of its domain, and apart from a normalization $c.$
So, what I want to find is the value of something like this
$$E(X^p) = c \int_a^b x^p \rho_N(x | \mu, \sigma) dx$$
where
$$ 1/c = \int_a^b \rho_N(x | \mu, \sigma) dx$$
Is there a known solution to this that I can refer to?