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.

It has two equivalent definitions. One is

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

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.

Derivatives

We may define the multivariate digamma function as

$\psi_p(a) = \frac{\partial \log\Gamma_p(a)}{\partial a} = \sum_{i=1}^p \psi(a+(1-i)/2) ,$

and the general polygamma function as

$\psi_p^{(n)}(a) = \frac{\partial^n \log\Gamma_p(a)}{\partial a^n} = \sum_{i=1}^p \psi^{(n)}(a+(1-i)/2).$

Calculation steps

• Since

$\Gamma_p(a) = \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+\frac{1-j}{2}),$

it follows that

$\frac{\partial \Gamma_p(a)}{\partial a} = \pi^{p(p-1)/4}\sum_{i=1}^p \frac{\partial\Gamma(a+\frac{1-i}{2})}{\partial a}\prod_{j=1, j\neq i}^p\Gamma(a+\frac{1-j}{2}).$

• By definition of the digamma function, ψ,

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

it follows that

$\frac{\partial \Gamma_p(a)}{\partial a} = \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+(1-j)/2) \sum_{i=1}^p \psi(a+(1-i)/2) = \Gamma_p(a)\sum_{i=1}^p \psi(a+(1-i)/2).$