exponential distribution and the gamma function

Question

I don't find the relationship between the expected value (theoretical) of an exponential distributed variable and the gamma function. I work on the paper Moments of the Log ACD model from Luc Bauwens(2008). And on page six they estimate the expected value form an exponential distributed variable with a gamma function.There is some further context, for example how the part with epsilon^(alpha*beta^j) stems from, but this is not the crucial part and certainly not why I struggle to understand the relationship to the equality of equation 3 in the picture. Alpha and beta are strict smaller than one.

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Answer 1

The PDF of $\epsilon \sim \exp(1)$ is $e^{-x}$ for $x \geq 0$. Hence $E[\epsilon^\gamma] = \int_0^\infty x^\gamma e^{-x}\,dx = \Gamma[1+\gamma]$ by the definition of the gamma function.