Multivariate gamma function

The multivariate gamma distribution , $\Gamma _{p}(\cdot )$, is a generalization of the gamma function that is useful in multivariate statistics.

It has two equivalent definitions:

$\Gamma _{p}(a)=\int _{S\in {\mathbf {S} }}\exp \left(-{\rm {trace}}(S)\right)\left|S\right|^{a-(p+1)/2}dS$

where ${\mathbf {S} }$ is the set of all positive-definite matrices. The other is more useful in practice:

$\Gamma _{p}(a)=\pi ^{p(p-1)/4}\Pi _{j=1}^{p}\Gamma \left[a+(1-j)/2\right].$

Thus

• $\Gamma _{1}(a)=\Gamma (a)$
• $\Gamma _{2}(a)=\pi ^{1/2}\Gamma (a)\Gamma (a-1/2)$
• $\Gamma _{3}(a)=\pi ^{3/2}\Gamma (a)\Gamma (a-1/2)\Gamma (a-1)$

and so on.