Multivariate gamma function

In mathematics, the multivariate Gamma function, Γ_p(·), is a generalization of the Gamma function. It is useful in multivariate statistics.

It has two equivalent definitions. One is

$\Gamma _{p}(a)=\int _{S>0}\exp \left(-{\rm {trace}}(S)\right)\left|S\right|^{a-(p+1)/2}dS$

where S>0 means S is positive-definite. The other one, more useful in practice, is

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

From this, we have the recursive relationships:

$\Gamma _{p}(a)=\pi ^{(p-1)/2}\Gamma (a)\Gamma _{p-1}(a-{\tfrac {1}{2}})=\pi ^{(p-1)/2}\Gamma _{p-1}(a)\Gamma [a+(1-p)/2]$

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.

References

• James, A. (1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". Annals of Mathematical Statistics. 35 (2): 475–501. doi:10.1214/aoms/1177703550. MR181057 Zbl 0121.36605.