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:

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

where ${\displaystyle |S|}$ denotes the determinant of ${\displaystyle S}$. The other one, more useful to obtain a numerical result is:

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

In both definitions, ${\displaystyle a}$ is a complex number whose real part satisfies ${\displaystyle \Re (a)>(p-1)/2}$. Note that ${\displaystyle \Gamma _{1}(a)}$ reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for ${\displaystyle p\geq 2}$:

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

Thus

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

and so on.

This can also be extended to non-integer values of ${\displaystyle p}$ 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 Wishart, Mahalanobis and others.

There also exists a version of the multivariate gamma function which instead of a single complex number takes a ${\displaystyle p}$-dimensional vector of complex numbers as its argument. It generalizes the above defined multivariate gamma function insofar as the latter is obtained by a particular choice of multivariate argument of the former.^[3]

Derivatives

We may define the multivariate digamma function as

${\displaystyle \psi _{p}(a)={\frac {\partial \log \Gamma _{p}(a)}{\partial a}}=\sum _{i=1}^{p}\psi (a+(1-i)/2),}$

and the general polygamma function as

${\displaystyle \psi _{p}^{(n)}(a)={\frac {\partial ^{n}\log \Gamma _{p}(a)}{\partial a^{n}}}=\sum _{i=1}^{p}\psi ^{(n)}(a+(1-i)/2).}$

Calculation steps

• Since

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

it follows that

${\displaystyle {\frac {\partial \Gamma _{p}(a)}{\partial a}}=\pi ^{p(p-1)/4}\sum _{i=1}^{p}{\frac {\partial \Gamma \left(a+{\frac {1-i}{2}}\right)}{\partial a}}\prod _{j=1,j\neq i}^{p}\Gamma \left(a+{\frac {1-j}{2}}\right).}$

• By definition of the digamma function, ψ,

${\displaystyle {\frac {\partial \Gamma (a+(1-i)/2)}{\partial a}}=\psi (a+(i-1)/2)\Gamma (a+(i-1)/2)}$

it follows that

${\displaystyle {\begin{aligned}{\frac {\partial \Gamma _{p}(a)}{\partial a}}&=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma (a+(1-j)/2)\sum _{i=1}^{p}\psi (a+(1-i)/2)\\[4pt]&=\Gamma _{p}(a)\sum _{i=1}^{p}\psi (a+(1-i)/2).\end{aligned}}}$