Digital Library of Mathematical Functions
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35 Functions of Matrix ArgumentProperties

§35.3 Multivariate Gamma and Beta Functions

Contents

§35.3(i) Definitions

35.3.1 Γm(a)=Ωetr(-X)|X|a-12(m+1)X,
(a)>12(m-1).
35.3.2 Γm(s1,,sm)=Ωetr(-X)|X|sm-12(m+1)j=1m-1|(X)j|sj-sj+1X,
sj, (sj)>12(j-1), j=1,,m.
35.3.3 Bm(a,b)=0<X<I|X|a-12(m+1)|I-X|b-12(m+1)X,
(a),(b)>12(m-1).

§35.3(ii) Properties

35.3.4 Γm(a)=πm(m-1)/4j=1mΓ(a-12(j-1)).
35.3.5 Γm(s1,,sm)=πm(m-1)/4j=1mΓ(sj-12(j-1)).
35.3.6 Γm(a,,a)=Γm(a).
35.3.7 Bm(a,b)=Γm(a)Γm(b)Γm(a+b).
35.3.8 Bm(a,b)=Ω|X|a-12(m+1)|I+X|-(a+b)X,
(a),(b)>12(m-1).