How to solve this integral involving exponential and exponential of exponential?

Question

I would like to find the moment generating function of $$ e^{-a-bX^2}, $$ where X is $N(\mu,\sigma^2)$.

This is equivalent to compute the following integral : $$ I = \int_{-\infty}^{+\infty}e^{te^{-a-bx^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}dx. $$

Any hint on how to integrate this? I guess it should involve substitution, but I don't see what could be efficient here.

Thanks a lot.

Answer 1

Notice that by symmetry of the Gaussian, the even order moments vanish.

Then by parts, borrowing an $x$ from $x^{2n}$,

$$M_{2n}=\int_{-\infty}^\infty x^{2n}e^{-x^2/2}dx=-\left.x^{2n-1}e^{-x^2/2}\right|_{-\infty}^\infty+(2n-1)\int_{-\infty}^\infty x^{2n-2}e^{-x^2/2}dx=(2n-1)M_{2n-2}$$ and by induction

$$M_{2n}=(2n-1)!!M_0.$$