Absolute moments of GH distribution

In this work, we focus on deriving the absolute moments of the general ized hyperbolic (GH) distribution via its mean-variance mixture structure. Conditioning on the GIG mixing variable reduces the problem to Gaussian moment calculations and known GIG power moments, avoiding direct integration with Bessel functions. The variance gamma (VG), normal inverse Gaussian (NIG), hyperbolic, Normal, and Student t distributions emerge as natural special or limiting cases of the generalized hyperbolic family. This approach offers a substantially simpler and more transparent derivation, facilitating both implementation and verification.