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.

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

(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.


We may define the multivariate digamma function as

and the general polygamma function as

Calculation steps[edit]

  • Since
it follows that
it follows that


  • James, A. (1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". Annals of Mathematical Statistics. 35 (2): 475–501. doi:10.1214/aoms/1177703550. MR 0181057. Zbl 0121.36605.
  • A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.