This is a question I encountered in my own research on Generalized Hyperbolic Secant (GHS) distributions. It is known that the Laplace transform of the basis measure for this family is $ $ L\left( \theta\right) =\left( \cos \theta\right) ^{-\sigma}$ $ where $ \sigma >0$ is the scale parameter and $ \vert \theta \vert < \pi/2$ .

Now I need to compute $ L^{\left( n\right) }\left( 0\right) =\left. \frac{d^{n}}{d\theta^{n}}L\left( \theta\right) \right\vert _{\theta=0}$ , i.e., the $ n$ th derivative of $ L\left( \theta\right) $ at $ 0$ for each even $ n$ . Note that $ L\left( \theta\right) $ is the Laplace transform of a density $ f$ that is symmetric around $ 0$ and $ $ L^{\left( n\right) }\left( 0\right) =\int_{-\infty}^{\infty}x^{n}f\left( x\right) dx. $ $ We know $ L^{\left( n\right) }\left( 0\right) =0$ if $ n$ is odd.

Since $ \cos^{\left( j\right) }\left( 0\right) =0$ whenever $ j$ is odd, in Faa di Bruno’s formula for $ L^{\left( n\right) }\left( 0\right) $ as long as $ \cos^{\left( j\right) }\left( \theta\right) $ with $ j$ odd appears, the corresponding summand is $ 0$ . But I was not able to single out the nonzero terms from this formula.

I also tried induction on $ L^{\left( n\right) }\left( 0\right) $ with $ n$ even but did not find a pattern yet. (Also tried integration by parts, but computing the primitive $ G$ of $ f$ and the primitive of $ G$ is much involved; also tried to directly compute $ \int_{-\infty}^{\infty}x^{n+2}f\left( x\right) dx$ with $ n$ even using contour integral (involving Gamma functions) but failed)

Any suggestions? Thank you.