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Is there an analytical solution for $\operatorname{E}[x^m\exp(kx)]$ when $k,m$ are positive integers and $x$ has skew normal distribution with parameters $\xi,\omega,\alpha$?

Wiki says that $\operatorname{E}[x]=\xi + \omega\delta\sqrt{\frac{2}{\pi}}$ where $\delta = \frac{\alpha}{\sqrt{1+\alpha^2}}$ and $\operatorname{E}[\exp(kx)]=2\exp\left(k\xi +\frac{k^2\omega^2}{2}\right)\Phi\left(k\omega\delta\right)$.

But how to find $\operatorname{E}[x^m\exp(kx)]$??? It involves rather complex integral over PDF. Wolfram Mathematica can't take the integral.

Ivan Balaban
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    Hint: differentiate $\operatorname{E}[\exp(kx)]$ with respect to $k$. – J.G. Jun 15 '23 at 19:25
  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jun 15 '23 at 20:08

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