Multivariate gamma function

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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.[1]

It has two equivalent definitions. One is given as the following integral over the positive-definite real matrices:

where denotes the determinant of . Note that reduces to the ordinary gamma function. The other one, more useful to obtain a numerical result is:

From this, we have the recursive relationships:


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.

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


We may define the multivariate digamma function as

and the general polygamma function as

Calculation steps[edit]

  • Since
it follows that
it follows that


  1. ^ James, Alan T. (June 1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". The Annals of Mathematical Statistics. 35 (2): 475–501. doi:10.1214/aoms/1177703550. ISSN 0003-4851.
  2. ^ Anderson, T W (1984). An Introduction to Multivariate Statistical Analysis. New York: John Wiley and Sons. pp. Ch. 7. ISBN 0-471-88987-3.