Multivariate gamma function

In mathematics, the multivariate gamma function Γ_p is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of the Wishart and inverse Wishart distributions, and the matrix variate beta distribution.[1]

It has two equivalent definitions. One is given as the following integral over the ${\displaystyle p\times p}$ positive-definite real matrices:

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

(note that ${\displaystyle \Gamma _{1}\left(a\right)}$ reduces to the ordinary gamma function). The other one, more useful to obtain a numerical result is:

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

From this, we have the recursive relationships:

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

Thus

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

and so on.

This can also be extended to non-integer values of p with the expression:

${\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}{\frac {G(a+{\frac {1}{2}})G(a+1)}{G(a+{\frac {1-p}{2}})G(a+1-{\frac {p}{2}})}}}$

Where G is the Barnes G-function, the indefinite product of the Gamma function.

The function is derived by Anderson[2] from first principles who also cites earlier work by Wishrt, Mahalabolis etc.