Multivariate gamma function

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

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

(note that $\Gamma _{1}\left(a\right)$ reduces to the ordinary gamma function). The other one, more useful to obtain a numerical result 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

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).

\Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma \left(a+{\frac {1-j}{2}}\right),

it follows that

{\frac {\partial \Gamma _{p}(a)}{\partial a}}=\pi ^{p(p-1)/4}\sum _{i=1}^{p}{\frac {\partial \Gamma \left(a+{\frac {1-i}{2}}\right)}{\partial a}}\prod _{j=1,j\neq i}^{p}\Gamma \left(a+{\frac {1-j}{2}}\right).

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

it follows that

{\begin{aligned}{\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)\\[4pt]&=\Gamma _{p}(a)\sum _{i=1}^{p}\psi (a+(1-i)/2).\end{aligned}}

James, A. (1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". Annals of Mathematical Statistics. 35 (2): 475&ndash, 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.

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