EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 4, 2017, 614-619
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Honorary Invited Paper

Selberg-Type Generalized Quadratic Forms Gamma
and Beta Integrals

Arjun K. Gupta1,∗, D. G. Kabe1,2

1 Bowling Green State University, Bowling Green, Ohio, USA
2 Deceased

Abstract. Although Selberg-type single positive definite symmetric matrices gamma and beta
integrals have been evaluated by several authors, see e.g., Askey and Richards [1], Gupta and Kabe
[2, 4], Mathai [8], and elsewhere in the vast multivariate statistical analysis literature. However,
several other types of Selberg-type integrals appear to have been neglected in the literature. Thus
e.g., Selberg-type integrals associated with inverse Wishart densities, inverse multivariate beta
densities, their noncentral counterparts, etc, have not been explored as yet. The present paper
records Selberg-type generalized quadratic forms gamma and beta integrals. Our methodology is
based on hypercomplex (HC) multivariate normal distribution theory, Kabe [6].

2010 Mathematics Subject Classifications: 62H10, 62H12

Key Words and Phrases: Selberg-Type Integral, Multivariate normal distribution, Hermitian
matrix, beta density

1. Introduction

The HC multivariate normal distribution is defined as follows. Let x1, x2 , ..., x4t,
t = 1

4 ,
1
2 , 1, 2 be 4t p× n real random matrices and for t = 2, i.e., the octonions case, set

Y = x1 + ix2,+jx3 + kx4 + lx5 +mx6 + nx7 + rx8, (1)

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

i2 = j2 = k2 = l2 = m2 = n2 = r2 = −1 = ijk = ilm = irn = jmr = kjr = knm. (2)

∗Corresponding author.

Email addresses: gupta@bgsu.edu (A. K. Gupta), Deceased (D. G. Kabe)

http://www.ejpam.com 614 c© 2017 EJPAM All rights reserved.



A. K. Gupta, D. G. Kabe / Eur. J. Pure Appl. Math, 10 (4) (2017), 614-619 615

For t = 1, t = 2, t = 4, t = 4i is the bioctonion case, Hypercomplex variables do not form a
field, they are known to form Clifford Algebras. The octonions conjugate of Y is defined
by

Ȳ = x1 − ix2 − jx3 − kx4 − lx5 −mx6 − nx7 − rx8. (3)

Note that Y Ȳ ′ is a positive definite HC Hermitian matrix (HCHM).
Next set∑

=
∑

1

+ i
∑

2

+ j
∑

3

+ k
∑

4

+ l
∑

5

+m
∑

6

+ n
∑

7

+ r
∑

8

, (4)

where
∑

1 is a p×p positive definite symmetric matrix, and
∑

2, ...,
∑

8 are real p×p skew

symmetric matrices. Note that
∑−1 =

∑
and

∑
is HCHM. Now setting dy = dx1...dx8,

Kabe [6] shows that the pn variate HC multivariate normal density of Y can be written as

f(Y ) = π−2pnt|
∑
|−2ntexp{−tr

∑
−1Y Ȳ }, (5)

and hence the HC Wishart density of the p× p HCHM G = Y Ȳ ′ is

f(G) = {Γp(2nt)}−1|
∑
|−2nt|G|−2t(n−p+1)−1exp{−tr

∑
−1G}, (6)

where
Γp(a) = πtp(p−1)Πp

i=1Γ(a− 2t(p− i)). (7)

Further for given two p× p HCHM matrices A and B, having HC Wishart densities with
n and q degrees of freedom, the density of the p× p HCHM R defined by

R = G−
1
2AG−

1
2 , A+B = G, (8)

is given by the expression

f(R) = {Bp(2nt, 2qt)}−1|I −R|2t(n−p+1)−1|R|2t(q−p+1)−1, (9)

where (see [5]),

Bp(a, b) =
Γp(a)Γp(b)

Γp(a+ b)
. (10)

If now ∧ is the p × p diagonal matrix of the roots of R, then Kabe [6, p.68, equation
(21)] shows that the Jacobian

J(R : ∧) = Πp
i<j(xi − xj)

4t,∧ = diag(λ1, ..., λp), (11)

and the HC multivariate beta density of ∧ is

f(∧) = {Bp(2nt, 2qt)−1|I − ∧|2t(n−p+1)−1| ∧ |2t(q−p+1)−1Πp
i<j(λi − λj)

4t, (12)

with a similar result for the density of the roots matrix of (6).
Now our paper proceeds as follows. The next section derives the real generalized

quadratic forms Wishart density (GQFWD), and section 3 develops the real generalized
quadratic forms multivariate beta density (GQFMBD). Section 4 records the gamma in-
tegrals, and section 5 records the beta integrals of the context.



A. K. Gupta, D. G. Kabe / Eur. J. Pure Appl. Math, 10 (4) (2017), 614-619 616

2. GQFWD

Let X be a p×n matrix of rank p ≤ n, and ∆ = diag(δ1, ..., δn) n×n diagonal matrix,
then GQFWD of p× p T is defined by the integral

f(T ) = k

∫
XX′=T

exp{−trX∆X ′}dX, (13)

where k, as a generic letter, denotes the normalizing constants of density functions in this
paper.

The moment generating function of T is

φ(θ) = k

∫
exp{−tr(X∆X ′ − θXX ′)}dX = Πn

i=1 |δiI − θ|−
1
2 , (14)

where θ, in the usual sense, is p× p positive definite symmetric matrix.
Indeed, Mathai [8, p.353], derives the density (13); however, his density function is not

suitable in our context. In our context inverting (14) we find that

f(T ) = |∆|−
1
2
p|δ1|−

1
2
pnexp{−δ1trT}|T |

1
2

(n−p−1){Γp(
1

2
n)}−1

.1F1(
1

2
(n− 1);

1

2
n; ((n− 1)δ1 − δ2 − ...− δn)T ) (15)

where δ1 = max(δ1, ..., δn). It is possible to use Mathai’s [8] GQFWD and GQFMBD
results to derive Selberg-type gamma and beta integrals of our context; however, our
expressions for the GQFWD and GQFMBD appear to be simpler than the ones given by
Mathai [8], and Mathai, Provost and Hayakawa [7, Chapter 5]

We proceed to prove (15) . Given the joint density of n gamma variates to be

f(y1, ..., yn) = kexp{−(α1y1 + + αnyn)}yg1−1
1 ...ygn−1

n , (16)

the density of t = y1 + ... + yn is desired, where α′s are distinct real positive constants.
The moment generating function Ψ(θ) of t is,

Ψ(θ) = (α1 − θ)−g1 ...(αn − θ)−gn

= (α1 − θ)−g1((α1 − θ)− (α1 − α2))−g2 ...((α1 − θ)− (α1 − αn))−gn

= (α1 − θ)−(g1+...+gn+r2+...rn)
∑∞

r2=0 ...
∑∞

rn=0

(
g2+r2−1

r2

)
...
(
gn+rn−1

rn

)
(α1 − α2)r2 ...

(α1 − αn)rn , (17)

where α1 = max{α1, ..., αn}. And now inverting (17) we have that

f(T ) = (Γ(g1 + + gn))−1Πn
i=1α

gj
j (α1)g1+...+gnexp{−α1t}t(g1+...+gn−1)



A. K. Gupta, D. G. Kabe / Eur. J. Pure Appl. Math, 10 (4) (2017), 614-619 617

ϕ(g2, ..., gn; g1 + ...+ gn; (α1 − α2)t, ..., (α1 − αn)t). (18)

Here, ϕ =
∑∞

r2=0 ...
∑∞

rn=0
Γ(g2+r2)...Γ(gn+rn)(α1−α2)r2 (α1−αn)rn

Γ(g1+...+gn+r2+...+rn)r2!...rn!

=1 F1(g2 + ...+ gn; g1 + ...+ gn; ((n− 1)α1 − α2 − ...− αn)t). (19)

To prove (19) we proceed as follows. The sum of the two noncentral p×p Wishart matrices
A and B, with n and q df, and noncentrality parameters ∆ and Ω, is again noncentral
Wishart with (n+q)df, and (∆+Ω) as noncentrality parameter. With 2g=(p+1), we write
this result as

k

∫
exp{−tr(A+B)}|A|n−g|B|q−g 0F1(n; ∆A)0F1(q; ΩB)dA dB

= exp{−trD}|D|n−g 0F1(n+ q; (∆ + Ω)D), (20)

0F1(n; ∆A)0F1(q; ΩB) =0 F1(n+ q; (∆ + Ω)(A+B)). (21)

Now Mathai [8, p. 339, Theorem 5.5] defines

ϕ(b1, b2; c;X1, X2)

=
∫
|U1|d1−g|U2|d2−g|I − U1 − U2|c−d1−d2−g 1F1(b1; d1;X1U1) 1F1(b2; d2;X2U2)dU1dU2

=
∫
|U1|d1−g|U2|d2−g|I − U1 − U2|c−d1−d2−gexp{−tr(Z1 + Z2)}
|Z1|b1−g|Z2|b2−g 0F1(d1;X1U1Z1)0F1(d2;X2U2Z2)dZ1dZ2dU1dU2

=
∫
|U1|d1−g|U2|d2−g|I − U1 − U2|c−d1−d2−gexp{−tr(Z1 + Z2)}
|Z1|b1−g|Z2|b2−g 0F1(d1 + d2; (X1 +X2)(U1 + U2)(Z1 + Z2))dZ1dZ2dU1dU2

=
∫
|U1|d1−g|U2|d2−g|I − U1 − U2|c−d1−d2−g

1F1(b1 + b2; d1 + d2; (X1 +X2)(U1 + U2))dU1dU2

= k2F2(d1 + d2; b1 + b2; d1 + d2; c;X1 +X2)

= k1 1F1(b1 + b2; c;X1 +X2),
(22)

and hence (15) and (19) follow. All matrices in (22) are p× p positive definite matrices.

3. GQFMBD

The GQFMBD of the p× p matrix M defined by the integral

f(M) =

∫
R
g(T1)g(T2)dT1dT2, (23)



A. K. Gupta, D. G. Kabe / Eur. J. Pure Appl. Math, 10 (4) (2017), 614-619 618

whereR = (T1 + T2)
1
2M(T1 + T2)

1
2 = T1. The joint density of p× p T1 and T2 is

f(T1, T2) = |∆|p(δ1)pnexp{−δ1tr(T1 + T2)}|T1|
1
2

(n−p−1)|T2|
1
2

(q−p−1)

1F1(
1

2
(n− 1);

1

2
n; δT1)1F1(

1

2
(q − 1);

1

2
q; δT2). (24)

Note that the matrix M has a doubly noncentral multivariate beta density derived by
Gupta and Kabe [3] as follows.

f(T1, T2) = k exp{−δ1tr(T1 + T2)}|T1|
1
2

(n−p−1)|T2|
1
2

(q−p−1)∫
exp{−tr(Z1 + Z2)}|Z1|

1
2

(n−1)−g|Z2|
1
2

(q−1)−g

0F1(
1

2
n; δT1, Z1) 0F1(

1

2
q; δT2)dZ1dZ2, (25)

and hence the doubly noncentral multivariate beta density of M is

f(M) = k exp{−tr(T1 + T2)}|Z1|
1
2

(n−1)−g|Z2|
1
2

(q−1)−g|M |
1
2

(n−p−1)

|I −M |
1
2

(q−p−1)
1F1(1

2(n+ q); 1
2n; δM(Z1 + Z2)dZ1dZ2

= |∆|p(δ1)
1
2

(n+q−2)|M |
1
2

(n−p−1)|I −M |
1
2

(q−p−1){Bp(1
2n; 1

2q)}
−1

2F1(
1

2
(n+ q − 2);

1

2
(n+ q);

1

2
n; δδ−p1 M). (26)

We now proceed to write the HC counterparts of (15) and (25), with δ = ((n− 1)δ1− δ2−
− δn).

4. HCGFWD

The HCGFWD of (15) is

f(T ) = |∆|2ptexp{−δ1trT}|T |2t(n−p+1)−1{Γp(2nt)}−1(δ1)2ptn
1F1(2t(n−1); 2tn; δT ), (27)

and hence the density of the roots matrix p× p ∧ is

f(∧) = |∆‖2ptexp{−δ1trT}|T |2t(n−p+1)−1{Γp(2nt)}−1(δ1)2ptn

1F1(2t(n− 1); 2tn; δT )Πp
i<j(λi − λj)

4t (28)

and setting (2t = h) we find that∫
exp{−δ1tr∧}| ∧ |g−1

1F1((n− 1)h;nh; δ∧)Πp
i<j(λi − λj)2hd∧

= |∆|−ph(δ1)pnh
Γp(g + hp− h)

Γ(p+ 1)
(29)

Note that in (28) the roots are ordered, but in Selberg-type they are unordered hence the
factor Γ(p+ 1) in (29).



REFERENCES 619

5. HCGQFMBD

The HCGQFMBD of p× p M is

f(M) = |∆|4pt(δ1)2pt(n−q+2){Bp(2nt; 2qt)}−1|M |2t(n−p+1)−1|I −M |2t(q−p+1)−1

2F1(2t(n+ q − 2); 2t(n+ q); 2tn; δ−4pt
1 δM),

f(∧) = {Bp(2nt; 2qt)}−1|∆|4pt(δ1)2pt(n−q+2)| ∧ |2t(n−p+1)−1|I − ∧|2t(q−p+1)−1

2F1(2t(n+ q − 2); 2t(n+ q); 2tn; δ−4pt
1 δ∧)Πp

i<j(λi − λj)4t

and hence setting 2t = h, we have that∫
| ∧ |g−1|I − ∧|t−1

2F1(h(n+ q − 2);h(n+ q);hn; δ(δ1)−2hp∧)Πp
i<j(λi − λj)4td∧

=
Bp(g + hp− h; t+ hp− h)|∆|−2hp(δ1)−hp(n−q+2)

Γ(p+ 1)
(30)

Thus (29) and (30) are the Selberg-type integrals of our context.

References

[1] Askey, Richard and Richards, Donald. (1989) Selbergs second beta integral and an
integral of Mehta, Probability, Statistics, and Mathematics. Academic Press, (Karlin
Volume), New York.

[2] Gupta, A.K. and Kabe, D.G. (2005). On Selberg beta integrals Random Obr. And
Stocastic Eqn 13, 11-16.

[3] Gupta, A.K. and Kabe, D.G. (2007). The doubly noncentral multivariate beta distri-
bution. J. Appli. Stat. Sc. 1,21-25.

[4] Gupta, A.K. and Kabe, D.G. (2008). Selberg-type squared matrices gamma and beta
integrals. Eu. J. Pure and Appli. Math. 1,197-201.

[5] Gupta, A.K. and Nagar, D.K. (2007). Matrix Variate Distributions

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

[7] Mathai, A.M., Hayakawa, T., Provost, S.B. (1995). Bilinear Forms and Zonal Poly-
nomials. Springer-Verlag, New York.

[8] Mathai, A.M. (1997). Jacobians of Matrix Transformations and Functions of Matrix
Arguments. World Scientific, London, England.