Multivariate gamma function
In mathematics, the multivariate gamma function Γp 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.
It has two equivalent definitions. One is given as the following integral over the positive-definite real matrices:
. The other one, more useful to obtain a numerical result is:
From this, we have the recursive relationships:
Thus
and so on.
This can also be extended to non-integer values of p with the expression:
Where G is the Barnes G-function, the indefinite product of the Gamma function.Derivatives
We may define the multivariate digamma function as
and the general polygamma function as
