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 p × p {\displaystyle p\times p} positive-definite real matrices:
Γ p ( a ) = ∫ S > 0 exp ( − t r ( S ) ) | S | a − p + 1 2 d S , {\displaystyle \Gamma _{p}(a)=\int _{S>0}\exp \left(-{\rm {tr}}(S)\right)\,\left|S\right|^{a-{\frac {p+1}{2}}}dS,}
where | S | {\displaystyle |S|} denotes the determinant of S {\displaystyle S} . The other one, more useful to obtain a numerical result is:
Γ p ( a ) = π p ( p − 1 ) / 4 ∏ j = 1 p Γ ( a + ( 1 − j ) / 2 ) . {\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma (a+(1-j)/2).}
In both definitions, a {\displaystyle a} is a complex number whose real part satisfies ℜ ( a ) > ( p − 1 ) / 2 {\displaystyle \Re (a)>(p-1)/2} . Note that Γ 1 ( a ) {\displaystyle \Gamma _{1}(a)} reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for p ≥ 2 {\displaystyle p\geq 2} :
Γ p ( a ) = π ( p − 1 ) / 2 Γ ( a ) Γ p − 1 ( a − 1 2 ) = π ( p − 1 ) / 2 Γ p − 1 ( a ) Γ ( a + ( 1 − p ) / 2 ) . {\displaystyle \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
Γ 2 ( a ) = π 1 / 2 Γ ( a ) Γ ( a − 1 / 2 ) {\displaystyle \Gamma _{2}(a)=\pi ^{1/2}\Gamma (a)\Gamma (a-1/2)}
Γ 3 ( a ) = π 3 / 2 Γ ( a ) Γ ( a − 1 / 2 ) Γ ( a − 1 ) {\displaystyle \Gamma _{3}(a)=\pi ^{3/2}\Gamma (a)\Gamma (a-1/2)\Gamma (a-1)}
and so on. This can also be extended to non-integer values of p {\displaystyle p} with the expression:
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