EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 1, No. 1, 2008, (197-201)
ISSN 1307-5543 – www.ejpam.com

Honorary Invited Paper

Selberg-type squared matrices gamma and beta integrals
A. K. Gupta1,∗, D. G. Kabe2

1 Bowling Green State University, Bowling Green, Ohio, 43403-0223 USA
2 5971 Greensboro Drive, Mississauga, ON, L5M 5S5 Canada

Abstract. Although Selberg-type, single positive definite symmetric matrices, gamma and beta integrals
are evaluated by several authors; see e.g., Askey and Richards (1989), Gupta and Kabe (2005), Mathai
(1997), and elsewhere in the vast multivariate statistical analysis literature; the Selberg-type squared ma-
trice gamma and beta interals appear to have been neglected. So also Selberg-type of integrals of positive
signature symmetric matrices, skew symmetric matrices are neglected in the literature. This paper records
Selberg-type squared matrices beta and gamma integrals.

Key words: Selberg-type integrals; squared matrices integrals; hypercomplex normal distributions;multivariate
gamma and beta densities.

1. Introduction

We first list a few Selberg-type beta and gamma readily available integrals. The following
three gamma integrals are listed by Mathai (1997, p. 231)∫

exp{−1
2
tr Λ2}

∏
i<j

(λi − λj)2hdΛ = π−
1
2
p(Γ(1 + h))−p

p∏
i=1

Γ(ih+ 1) (1)

where Λ = diag(λ1, . . . , λp), 0 < λi <∞,∫
exp{−tr; Λ}|Λ|g−1

∏
i<j

(λi− λj)2hdΛ =

[
p∏
i=1

Γ(g + (i− 1)h)Γ(ih+ 1)

]
[Γ(1 + h)]−p. (2)

and∫
exp{−tr Λ2}

∏
i<j

(λ2
i − λ2

j )
2hdΛ = [Γ(1 + h)]−p

p∏
i=1

γ

(
1
2

+ (i− 1)h
)

Γ(ih+ 1), (3)

which follows from Mathai (1997, p. 114, 2.1.4).

∗Corresponding author. Email address: gupta@math.bgsu.edu ( A. Gupta )

http://www.ejpam.com 197 c© 2007 EJPAM All rights reserved.



A.Gupta, D. Kabe / Eur. J. Pure Appl. Math, 1 (2008), (197-201) 198

Askey and Richards (1989), Gupta and Kabe (2005) record∫
|Λ|g−1|I −Λ|t−1

∏
i<j

(λi−λj)2hdΛ = [Γ(1 +h)]−p
p∏
i=1

Γ(g + (i− 1)h)Γ(t+ (i− 1)h)
Γ(g + t+ (i− h))

(4)

and∫
|Λ|g−1(1− trΛ)t−1

∏
i<j

(λi − λj)2hdΛ =
[Γ(1 + h)]−pΓ(t)π

1
2
hp(p−1)

Γ(t+ pg + p(p− 1)h)
p∏
i=1

Γ(g + (i− 1)h)Γ(ih+ 1)

.

(5)
Setting

Γp(a) = Πtp(p−1)
p∏
i=1

Γ(a− 2t(p− i)), (6)

we write∫
|I + Λ|−2t(n+q)|Λ|2t(n−p+1)−1

∏
i<j

(λi−λj)4tdΛ = Bp(2nt, 2qt)(Γ(1 + 2t))−p
p∏
i=1

Γ(2ti+ 1),

(7)
where Bp(a, b) = Γ0(a)Γp(b)

Γp(a+b) .
The present paper evaluates (1), and∫
|I + Λ2|−α

∏
i<j

(λi − λj)4tdΛ = 22tpπtp(p+1)Γp(α−
t

2
(p+ 1))[Γ(1 + 2t)]−p

p∏
i=1

Γ(2ti+ 1).

(8)
The integral (8) is listed by Mathai (1997, p. 112, 2.1.4).

Mathai (1997, p. 231, 4.1.2) lists a positive signature symmetric matrix Selberg-type gamma
integral. The positive signature symmetric matrice Selberg-type beta integrals follow on the same
lines. Mathai (1997) also lists several skew symmetric matrices integrals, but not Selberg-type
skew symmetric matrices integrals.

We state some useful results in the next section, and evaluate (1) and (8) in section 3. The
multivariate statisticsl analysis has now become extremely vast, and search for Selberg-type skew
symmetric matrix integrals is formidable.

2. Some Useful Results

Kabe (1984) develops the hypercomplex (HC) multivariate analysis distribution theory, and
we record here some relevant results.

Let X = (X ′1, . . . , X
′
4t)
′, t = 1

4 ,
1
2 , 1, 2 be 4t (p × n) real random matrices having the 4nt

variate normal density

g(X) = (4πt)−2nt|Σ0|−
1
2 exp

{
−trΣ−1

0 XX ′

4t

}
, (9)



A.Gupta, D. Kabe / Eur. J. Pure Appl. Math, 1 (2008), (197-201) 199

where Σ0, p× p, is

[Σ1 − Σ2 − Σ3 − Σ4 − Σ5 − Σ6 − Σ7 − Σ8Σ2Σ1 − Σ4Σ3 − Σ6Σ5Σ8 − Σ7Σ3Σ4Σ1

-Σ2 −Σ7Σ8Σ5Σ4Σ4 −Σ3Σ2Σ1 −Σ8Σ7 −Σ4Σ3Σ5Σ6Σ7Σ8Σ1 −Σ2 −Σ3 −Σ4Σ6 −Σ5 −
Σ8 − Σ7Σ2Σ1Σ4Σ3Σ7 − Σ8 − Σ5Σ6Σ3 − Σ4Σ1Σ2 − Σ8Σ7Σ6 − Σ5Σ4Σ3 − Σ2Σ1], (10)

where now t = 2, and we use Hamilton’s octonions, Halberstam and Ingram (1967, p. 654,
equation (1)). Similar HC theory follows by using Cayley’s or Young’s octonions or bioctonions.

In the context octonion case Σ1 is a real positive definite symmetric matrix, and Σ2, . . . ,Σ8

are real p× p skew symmetric matrices. Now we set

Y = X1 + iX2 + jX3 + kX4 + `X5 +mX6 + nX7 + rX8, (11)

where i, j, k, `,m, n, r octonions satisfy the multiplication rule

i2 = j2 = k2 = `2 = m2 = n2 = r2 = 1 = ijk = i`n = irn = j`n = jmn = kir = knm,
(12)

and observe that

Ȳ = X1 − iX2 − jX3 − kX4 − `X5 −mX6 − nX7 − rX8, (13)

is the HC conjugate of Y , and
dY = dX1dX2 . . . dX8. (14)

After some formidable algebra, Kabe (1984) writes the pn variate normal density of Y to be

g(Y ) = π−2pnt|Σ|−2nt exp{−trΣ−1Y Ȳ ′}, (15)

where
Σ = Σ1 + iΣ2 + jΣ3 + kΣ4 + `Σ5 +mΣ6 + nΣ7 + rΣ8, (16)

is the HC Hermitian p× p matrix. All of its roots are real and positive. There are also positive sig-
nature HC symmetric matrices, HC skew symmetric matrices, and they can be used to generalize
or evaluate integrals of type Mathai (1997, p. 231, 4.1.2).

On setting Y Ȳ ′ = G, the HC Hermitian p × p Wishart matrix, Kabe (1984, p. 67, equation
(14)) records this Wishart density

g(G) = (Γp(2nt))−1|Σ|−2nt|G|2t(n−p−1)−1 exp{−trΣ−1G}, (17)

a gamma-type HC Hermitian p× p matrix density. The first kind beta density is

g(G) = [Bp(2nt, 2qt)]−1|I −G|2t(n−p−1)−1|G|2t(q−p−1)−1, (18)

and the second kind beta density is

g(G) = [Bp(2nt, 2qt)]−1|I +G|−2t(n+q)|G|2t(n−p−1)−1. (19)



A.Gupta, D. Kabe / Eur. J. Pure Appl. Math, 1 (2008), (197-201) 200

3. Selberg Squared Matrices Integrals

It follows from Mathai, Provost, and Hayakawa (1995, p. 200, 4.3.2) that∫
exp{−tr G2}dG = 22tpπtp(p+1). (20)

Kabe (1984, p. 68) shows that G = OΛO′, where O is HC unitary, the Jacogian of transformation
from G to Λ is given by

J(G; Λ) =
∏
i<j

(λi − λj)4t. (21)

From (20) we conclude that∫
exp{−trΛ2}

∏
i<j

(λi − λj)4tdΛ = 22tpπtp(p+1)(Γ(1 + 2t))−p
p∏
i=1

Γ(2ti+ 1). (22)

Now we note that∫
|I +G2|−αdG =

∫ ∫
exp{−tr(I +G2)Z}|Z|α−

1
2

(p+1)dZdG. (23)

Setting G = Z−
1
4BΣ−

1
4 , and integrating with respect to Z first and then with respect to B the

integral (23) is written as∫
exp{−tr Z − tr B2}dB|Z|α−

1
4

(p+1)− 1
2

(p+1)dZ = 22tpπtp(p+1)Γp

(
α− t

2
(p+ 1)

)
, (24)

where t = 1
4 . When G is HC Hermitian we set in (24) t = t.

It follows from (24) that∫
|I +G2|−αdG = 22tpπtp(p+1)Γp

(
α− t

2
(p+ 1)

)
, (25)

and from (25) that∫
|I + Λ2|−α

∏
i<j

(λi−λj)4tdΛ = 22tpπtp(p+1)Γp

(
α− t

2
(p+ 1)

)
[Γ(1 + 2t)]−p

p∏
i=1

Γ(2ti+ 1).

REFERENCES

1. Askey, Richard and Richards, Donald. Selberg’s second beta integral and an integral of Mehta,
in Probability, Statistics, and Mathematics. Academic Press, New York, (Karlin Volume), (1989).

2. Halberstam, H. and Ingram, R. E. The Mathematical Papers of Sir William Hamilton, Vol. III,
Cambridge University Press, London (1967).



A.Gupta, D. Kabe / Eur. J. Pure Appl. Math, 1 (2008), (197-201) 201

3. Gupta, A. K. and Kabe, D. G. On Selberg’s beta integrals. Random Oper. and Stoch. Eqn. 13,
11-16, 2005.

4. Kabe, D. G. Classical statistical analysis based on a certain hypercomplex multivariate normal
distribution. Metrika 31, 63-76, (1984).

5. Mathai, A. M., Provost, S., and Hayakawa, T. Bilinear Forms and Zonal Polynomials, Springer-
Verlag, New York, (1995).

6. Mathai, A. M. Jacobians of Matrix Transformations. World Scientific, London, (1997).