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.

positive-definite real matrices:

denotes the determinant. The other one, more useful to obtain a numerical result is:

is a complex number:

Thus

and so on.

with the expression:

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

The function is derived by Anderson from first principles who also cites earlier work by Wishart, Mahalanobis and others.

-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.

Derivatives

We may define the multivariate digamma function as

and the general polygamma function as

Calculation steps

• Since

• it follows that

• By definition of the digamma function, ,

• it follows that

References

• 1.

• 2. A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.