Expressing integral as gamma function

Question

I was trying to compute the expected value $E[X^k]$ for a Weibull distribution, and I encountered this integral $$\int_{-\infty}^\infty t^{b+k-1}e^{-ct^b}dt$$ where $k>0$ is an integer and $c>0$. Can I somehow express it in terms of the gamma function $$\Gamma(a)=\int_0^\infty y^{a-1}e^{-y}dy$$for some $a>0$?

Answer 1

For the parameters where it's defined, the integral $\int_0^\infty$ is known as moments of the higher exponential function, which indeed is essentially a function of the Gamma function.

See

Wikipedia: Stretched_exponential_function#Mathematical_properties

You can start at the result and read off the parameter substitution.