Expectation of power function in Gamma Distribution?

Question

If $X$~Gamma($\alpha,\beta$), does the expectation of power function, $E[X^t], t>0$, have a closed-form solution? I know that when $t$ is a positive integer (natural number), $E[X^t]$ can be calculated by using the moment generating function $M_X(t)$. I have a question when t is positive real number, $t\in R_+$.

Answer 1

You do not need to use any moment generating function. Recall that $$ \Gamma_{\alpha,\beta}(x)=\frac{1}{\Gamma(\alpha)\beta^{\alpha}}x^{\alpha-1}e^{-x/\beta} $$ So you multiply $X^{k}$ on the front just change the kernel to a different set of value, say $(\alpha+k, \beta)$. The rest you can derive yourself by taking the quotient.