Bessel-type functions of matrix variables

35 (2005), 31–45

Bessel-type functions of matrix variables

Salem Ben Sai¨d

(Received November 18, 2003) (Revised October 18, 2004)

Abstract. In the present work we compute explicitly a certain type of hypergeometric functions of matrix variables given as an integral of a Gaussian-type kernel. In the case of one variable, these functions are related to the modified Bessel function of the third kind.

1. Introduction

This paper deals with explicit computations of certain type of hyper- geometric functions related to the linear groups Uðp;qÞ and Spð2n;RÞ. In doing this, some integral formulas over the group of unitary matrices are given. To be more precise, let us take the case of Uðp;qÞ.

For p;qAN and n¼ pþq, let Ip;q:¼ Ip 0 0 Iq

be the diagonal matrix with pcopies of (þ1) and qcopies of (1) along the diagonal. DefineUðp;qÞ as the set of invertible matrices gAMðn;CÞ such that gI_p;qg¼I_p;q, where g:¼g^t.

For diagonal matrices a:¼diagða1;. . .;apÞ and b:¼diagðb₁;. . .;b_qÞ, such that a_iþb_j00, we define

z_p;qða;bÞ:¼ ð

Uðp;qÞ

e^tr _{a 0}

0 b

ðggÞ¹ dg:

Here ‘‘tr’’ means the usual trace of a matrix. If p¼q¼1, we can easily show that

z_1;1ða;bÞ ¼c₀ðaþbÞ¹⁼²K₁₌₂ðaþbÞ; where KnðzÞ is the modified Bessel function of the third kind

K_nðzÞ ¼ ffiffiffiffiffi

p 2z r e^z=2

G n þ¹₂ ðy

0

e^ttⁿ¹⁼² 1þt z n1=2

dt

2000 Mathematics Subject Classification. 11T35, 65D20.

Key Words and phrases. Matrices and determinant, Computation of special matrix functions, Integrals over compact groups.

for Renþ¹₂

>0 and jargzj<p. As we can see, the function z_p;q is a multivariate analogue of the modified Bessel function K_n. To compute z_p;q, the main idea is to write z_p;q as an integral over the unit ball Dp;q:¼ fzAMðp;q;CÞ jdetðIpzzÞ>0g, and to use the polar decomposition of Dp;q. In doing this, we also obtain the explicit formula of

0F0ðS;TÞ:¼ ð

UðmÞ

e^trðuSuTÞdu

forS¼diagðs1;. . .;s_m1;0;. . .;0ÞandT¼diagðt1;. . .;t_m2;0;. . .;0Þ. HereUðmÞ denotes the set of unitary matrices uAMðm;CÞ. It turns out that 0F0ðS;TÞ was introduced by A. T. James in [James, 1964] as a generalization of the usual hypergeometric function 0F0ðSÞ ¼e^trðSÞ.

The Bessel-type functions under investigation play a crucial role in the theory of random matrices, mainly when one needs to derive explicit formulas for the correlation functions of the random variables (see for instance [Bre´ zin- Hikami, 2001, Bre´ zin-Hikami, 2003]).

The following notations will be used through out the paper. For a matrix xwe write x¼x^t where x^t is the transpose of x. Ifx₁;x₂;. . .;x_r are complex numbers, diagðx1;x2;. . .;xrÞ

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

rr

denotes the diagonal matrix of size rr.

If x and y are two square matrices of size rr and ss, respectively, exp½trðxþyÞ stands for exp½trðxÞexp½trðyÞ where ‘‘exp’’ is the exponential function. For r;sAN, the element I_r;s is the diagonal matrix diag½Ir;Is, where IN is the NN identity matrix. For rAN, Sr denotes the group of permutations.

2. The Uðp;qÞ-case

Let p;qAN, and assume that qbp. We define

Uðp;qÞ ¼ fgAGLðn;CÞ jgIp;qg¼Ip;qg ðn¼pþqÞ;

where GLðn;CÞ denotes the set of nn-invertible matrices. For g¼ A B

C D

AUðp;qÞ, the defining condition of Uðp;qÞ implies the following relations

ðaÞ AABB ¼Ip ðeÞ C¼DBA¹; ðbÞ CCDD¼ Iq ðfÞ B¼ACD¹; ðcÞ AACC¼I_p ðgÞ C¼D¹BA;

ðdÞ BBDD¼ Iq ðhÞ B¼A¹CD:

For all a¼diagða1;. . .;apÞ with ai>0, and b ¼diagðb₁;. . .;b_qÞ with b_i>0, let

z_p;qða;bÞ ¼ ð

Uðp;qÞ

exp½trðdiag½a;bðggÞ¹Þdg:

For g¼ A B C D

,

diag½a;bðggÞ¹¼ aðAAþBBÞ aðACBDÞ bðCADBÞ bðCCþDDÞ

:

Therefore, by the relations (a) and (b) we have

trðdiag½a;bðggÞ¹Þ ¼trðaðAAþBBÞ þbðCCþDDÞÞ

¼trðað2AAIpÞ þbð2DDIqÞÞ:

Let D_p;q be the domain defined by

Dp;q¼ fTAMðp;q;CÞ jdetðIpTTÞ>0g:

The measuredmðTÞ ¼detðIpTTÞ^pqdT is theUðp;qÞ-invariant measure on D_p;q where dT is the Lebesgue measure on D_p;q.

The map Uðp;qÞ !Dp;q defined by g¼ A B

C D

7!T ¼BD¹

is an isomorphism. Using the relations ðaÞ;. . .;ðgÞ and (h), we can write AA¼ ðIpTTÞ¹ and DD ¼ ðIqTTÞ¹.

Next, we write UðNÞ1UðN;0Þ. It is well known that for all functions F defined on Uðp;qÞ, such that FðgkÞ ¼FðgÞ for all kA UðpÞ 0

0 UðqÞ

, there exists a function F^#:Dp;q!C defined by F^#ðTÞ ¼FðgÞ such that

ð

Uðp;qÞ

FðgÞdg¼ ð

Dp;q

F^#ðTÞdmðTÞ:

Therefore, if FðgÞ ¼exp½trðdiag½a;bðggÞ¹Þ, there exists a complex valued function F^# on Dp;q such that

F^#ðTÞ ¼exp½trða½2ðIpTTÞ¹Ip þb½2ðIqTTÞ¹IqÞ

¼exp½trðaðIpTTÞ¹ðIpþTTÞ þbðIqTTÞ¹ðIqþTTÞÞ

¼exp½trðaþbþ2aðIpTTÞ¹TTþ2bðIqTTÞ¹TTÞ:

By [Hua, 1963], forT ADp;q, there exists uAUðpÞ andvAUðqÞ such that T ¼uLv, where

L¼

l1 0 0 0

.. .

... .. .

...

0 lp 0 0

2 66 4

3 77

5AMðp;q;RÞ

and 1>l1bl2b blpb0. Hence TT¼udiag½l²₁;. . .;l²_p

|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

pp

u; TT¼vdiag½l₁²;. . .;l_p²;0;. . .;0

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

qq

v:

Therefore F^# can be written in terms of u, v and L as

F^#ðTÞ ¼exp½trðaþbÞexpð2 trðu¹auðIpLLÞ¹LLÞÞ expð2 trðv¹bvðIqLLÞ¹LLÞÞ:

Consider the map c:D_p;q!1 taking each T AD_p;q to the collection of the eigenvalues of ffiffiffiffiffiffiffiffiffiffi

TT

p . The image of the Lebesgue measure dT on Dp;q

with respect to the map c is the measure on 1 given by

c Y

1ai<jap

ðl_i²l_j²Þ²Y^p

i¼1

l_i^2ðqpÞþ1dli;

for some constant c. Thus, the image of the measure dmðTÞ ¼ detðIpTTÞ^pqdT is

c Y

1ai<jap

ðl_i²l_j²Þ²Y^p

i¼1

l^2ðqpÞþ1_i ð1l_i²Þ^pqdl_i: ð2:1Þ

Hence, the function z_p;qða;bÞ is given by z_p;qða;bÞ ¼cexp½trðaþbÞ

ð

UðpÞ

ð

UðqÞ

ð

1

expð2 trðu¹auðIpLLÞ¹LLÞÞ expð2 trðv¹bvðIqLLÞ¹LLÞÞ

Y

1ai<jap

ðl_i²l_j²Þ²Y^p

i¼1

l_i^2ðqpÞþ1ð1l²_iÞ^pqY^p

i¼1

dl_idudv:

Let

A:¼2ðIpLLÞ¹LL¼diag 2l₁²

1l₁²;. . .; 2l_p² 1l_p²

" #

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

pp

ð2:2Þ ;

and

B:¼2ðIqLLÞ¹LL¼diag 2l₁²

1l₁²;. . .; 2l_p²

1l_p²;0;. . .;0

" #

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

qq

ð2:3Þ :

It will be convenient for us to define new coordinates x_i¼ 2l_i²

ð1l_i²Þ. Then the set 1¼ fLj1>l1bl2b blpb0g becomes the set

X:¼ fdiagðx1;x2;. . .;xpÞ jx1bx2b bxpb0g.

The measure (2.1) in the coordinates x_i has the form

c Y

1ai<jaj

ðxix_jÞ²Y^p

i¼1

x_i^qpdx_i;

and the function z_p;qða;bÞ can be written as z_p;qða;bÞ ¼cexp½trðaþbÞ

ð

UðpÞ

ð

UðqÞ

ð

X

expðtrðu¹auAÞÞexpðtrðv¹bvBÞÞ

Y

1ai<jap

ðxix_jÞ²Y^p

i¼1

x_i^qpdx_idudv;

where A and B are given by (2.2) and (2.3).

Now we turn our attention to the integral formula over UðpÞ and UðqÞ.

For this we need to introduce some terminology.

For a multi-parameter t¼ ðt1;t₂;. . .;t_NÞ, the Vandermonde polynomial is defined by DðtÞ ¼ Q

1ai<jaN

ðtitjÞ. Let l¼ ðl1;. . .;lNÞAN^N. The Schur

polynomial S_lðt1;. . .;tNÞ is defined by

Slðt1;. . .;tNÞ ¼detðt_i^ljþNjÞ_1ai;jaN

DðtÞ :

For more details on Schur polynomials, we refer to [Macdonald, 1979, Chapter I]. We also need the following lemma.

Lemma 2.1 (cf. [Hua, 1963], Theorem 1.2.1). Let the power series

f_iðyÞ ¼X^y

k¼0

a^ðiÞ_k y^k:

Then for all ðy1;y2;. . .;yNÞ the following identity holds detðf_iðy_jÞÞ_1ai;jaN ¼ X

l1>l2>>lNb0

detða^ðiÞ_ljÞ_1ai;jaN detðy^l_i^jÞ_1ai;jaN; where l1;l2;. . .;lN are integers.

Now we are in position to compute the integral over the compact groups UðpÞ and UðqÞ.

Proposition2.2. (i)For A¼diagðx1;. . .;xpÞ, and fora¼diagða1;. . .;apÞ, we have

ð

UðpÞ

exp½trðu¹auAÞdu¼ ð1Þ^pðp1Þ=2Y^p

i¼1

ði1Þ! detðe^xiajÞ_1ai;jap Y

1ai<jap

ðxixjÞðaiajÞ:

(ii) For B¼diagðx1;x2;. . .;xp;0;. . .;0Þ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

qq

, and for b ¼diagðb₁;b₂;. . .;b_qÞ, we have

ð

UðqÞ

exp½trðv¹bvBÞdv

¼

ð1Þ^qðq1Þ=2Y^q

i¼1

ði1Þ! Y

1ai<jap

ðxix_jÞ Y

1ai<jaq

ðb_ib_jÞ

X

l1>l2>>lpb0

detðx^l_i^jÞ_1ai;jap Y^p

j¼1

ðljþqpÞ!

ðb₁Þ^l1þqp ðb_qÞ^l1þqp ...

... ðb₁Þ^lpþqp ðb_qÞ^lpþqp ðb₁Þ^qp1 ðb_qÞ^qp1

...

... ðb₁Þ ðb_qÞ

1 1

where l1;. . .;lp are integers.

Proof. (i) First, let us write the Taylor series of exp½trðu¹auAÞ as a series of Schur polynomials Sl, in the form

exp½trðu¹auAÞ ¼ X

l1bblpb0

d_l d!

ðlþdÞ!S_lð}ðu¹auAÞÞ;

where l¼ ðl1;. . .;lpÞ,d¼ ðp1;p2;. . .;0Þ,dl¼DðlþdÞ

DðdÞ , and}ðgÞstands for the collection ðz1;. . .;zpÞ of the eigenvalues of g. Therefore,

Iða;AÞ1 ð

UðpÞ

exp½trðu¹auAÞdu

¼ X

l1bblpb0

dl

d!

ðlþdÞ! ð

UðpÞ

Slð}ðu¹auAÞÞdu

¼ X

l1bblpb0

d!

ðlþdÞ!S_lðaÞS_lðAÞ:

To obtain the latter equality, we used the following well known functional equation

ð

UðpÞ

w_lðxuyu¹Þdu¼ 1

d_lw_lðxÞw_lðyÞ

where w_l is the central function on UðpÞ whose restriction to the set of di- agonal matrices in UðpÞ is equal to S_l (see for instance [Macdonald, 1979, Chapter I]).

Using the determinant formula of Sl, we deduce

Iða;AÞ ¼ d!

DðaÞDðAÞ X

l1bblpb0

detða^l_i^jþpjÞ_1ai;japdetðx^l_i^jþpjÞ_1ai;jap ðl1þp1Þ!ðl2þp2Þ! . . .lp!

¼ d!

DðaÞDðAÞ X

l1>>l_pb0

detða^l_i^jÞ_1ai;japdetðx^l_i^jÞ_1ai;jap

l1!l2! . . .lp! :

Let

fiðaÞ ¼e^xia ¼X^y

k¼0

ðxiÞ^k k! a^k: Using Lemma 2.1 where a^ðiÞ_k ¼ðxiÞ^k

k! , we obtain detðe^xiajÞ_1ai;jap¼detðfiðajÞÞ_1ai;jap

¼ X

l1>>lpb0

det ðxiÞ^lj lj!

!

1ai;jap

detða^l_i^jÞ_1ai;jap: Therefore, statement (i) holds.

(ii) Let b ¼diagðb₁;. . .;b_qÞ and let X ¼diagðx1;. . .;xp;xpþ1;. . .;xqÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

qq

. Using statement (i), we have

ð

UðqÞ

exp½trðv¹bvXÞdv¼c_q detðe^xibjÞ_1ai;jaq Y

1ai<jaq

ðxixjÞ Y

1ai<jaq

ðb_ib_jÞ;

where c_q ¼ ð1Þ^qðq1Þ=2Y^q

i¼1

ði1Þ!. Also, from the proof of statement (i), we have

ð2:4Þ

detðe^xibjÞ_1ai;jaq Y

1ai<jaq

ðxixjÞ ¼ X

l1>>lq1>lqb0

Y^q

j¼1

1 lj!

detðx^l_i^jÞ_1ai;jaq Y

1ai<jaq

ðxixjÞ detððb_iÞ^ljÞ_1ai;jaq: Now we set x_q¼0 in (2.4). Then all terms with lq>0 vanish, and we get

ð2:4Þ_jxq¼0

¼ X

l1>>lq1>0

Y^q1

j¼1

1 lj!

detðx^l_i^jÞ_1ai;jaq1 Y

1ai<jaq1

ðxix_jÞY^q1

i¼1

x_i

ðb₁Þ^l1 ðb_qÞ^l1

ðb₁Þ^lq1 ðb_qÞ^lq1

1 1

¼ X

l1>>lq1>0

Y^q1

j¼1

1 lj!

detðx^l_i^j1Þ_1ai;jaq1 Y

1ai<jaq1

ðxix_jÞ

ðb₁Þ^l1 ðb_qÞ^l1

ðb₁Þ^lq1 ðb_qÞ^lq1

1 1

:

After substituting li by liþ1, we obtain ð2:4Þ_jxq¼0

¼ X

l1>>lq1b0

Y^q1

j¼1

1 ðljþ1Þ!

detðx^l_i^jÞ_1ai;jaq1 Y

1ai<jaq1

ðxixjÞ

ðb₁Þ^l1þ1 b^l_q^1þ1

ðb₁Þ^lq1þ1 b^l_q^q1þ1

1 1

:

Setting now xq1¼0 and repeating this processðqp1Þ-times, we arrive at the following sum: if x_q¼0;xq1¼0;. . .;xpþ1¼0, then

ð

UðqÞ

exp½trðv¹bvXÞdvjxq¼¼xpþ1¼0

¼ cq

Y

1ai<jaq

ðb_ib_jÞ X

l1>l2>>lpb0

Y^p

j¼1

1 ðljþqpÞ!

detðx^l_i^jÞ_1ai;jap Y

1ai<jap

ðxixjÞ

ðb₁Þ^l1þqp ðb_qÞ^l1þqp

ðb₁Þ^lpþqp ðb_qÞ^lpþqp ðb₁Þ^qp1 ðb_qÞ^qp1

ðb₁Þ ðb_qÞ

1 1

: 9

(After the work on this paper was completed, we learned that the ar- gument presented above for statement (i) was used earlier by G. Olshanski and A. M. Vershik in [Olshanski-Vershik, 1996].)

Remark 2.3. The first statement of Proposition 2.2 can be proved in a number of di¤erent ways. For instance, it can be obtained by using the Harish-Chandra integral formula (some times also called HIZ integral) [Harish-Chandra, 1957], [Gross-Richards, 1989]. Another interesting way is to obtain the integral formula over UðpÞ from the spherical function on GLðp;CÞ by a passage to the limit. For more about the latest way described above, and in a general setting of compact Lie groups, we refer to [Ben Sai¨d- Ørsted, 2003].

Next we turn to the computation ofz_p;qða;bÞ. The proof of the following lemma is obvious.

Lemma 2.4. Let m be a measure on R. Then ð

R^N

detffkðx_lÞg_k;ldetfgkðx_lÞg_k;ldmðx1Þ. . .dmðxNÞ

¼N!det ð

R

fkðxÞgmðxÞdmðxÞ

k;m

;

whenever the right-hand side of the equation makes sense.

Using Proposition 2.2, the function z_p;qða;bÞ is given by z_p;qða;bÞ ¼ cexp½trðaþbÞ

Y

1ai<jap

ðaia_jÞ Y

1ai<jaq

ðb_ib_jÞ

X

l1>>lpb0

Y^p

j¼1

1 ðljþqpÞ!

ðb₁Þ^l1þqp ðb_qÞ^l1þqp

ðb₁Þ^lpþqp ðb_qÞ^lpþqp ðb₁Þ^qp1 ðb_qÞ^qp1

ðb₁Þ ðb_qÞ

1 1

ð

X

detðe^xiajÞ_1ai;jap detðx^l_i^jÞ_1ai;jap Y

1ai<jap

ðxix_jÞ²

Y

1ai<jap

ðxixjÞ²Y^p

i¼1

x_i^qpdxi:

Using Lemma 2.4, we deduce that ð

X

detðe^xiajÞ_1ai;japdetðx^l_i^jÞ_1ai;jap Y

1ai<jap

ðxixjÞ²

Y

1ai<jap

ðxix_jÞ²Y^p

i¼1

x_i^qp dx_i

¼cdet ð^y

0

e^xaix^ljþqpdx

1ai;jap

¼cdet Gðljþqpþ1Þ a^l_i^jþqpþ1

!

1ai;jap

¼cdet ðljþqpÞ!

a^l_i^jþqpþ1

!

1ai;jap

;

where c is a constant. Therefore z_p;qða;bÞ ¼ cexp½trðaþbÞ

Y

1ai<jap

ðaiajÞ Y

1ai<jaq

ðb_ib_jÞ X

l1>>lpb0

Y^p

j¼1

1 ðljþqpÞ!

ðb₁Þ^l1þqp ðb_qÞ^l1þqp

ðb₁Þ^lpþqp ðb_qÞ^lpþqp ðb₁Þ^qp1 ðb_qÞ^qpþ1

ðb₁Þ ðb_qÞ

1 1

det ðljþqpÞ!

a^l_i^jþqpþ1

!

1ai;jap

¼ cexp½trðaþbÞ Y

1ai<jap

ðaia_jÞ Y

1ai<jaq

ðb_ib_jÞY^p

i¼1

a_i^qpþ1

X

l1>l2>>lpb0

ðb₁Þ^l1þqp ðb_qÞ^l1þqp

ðb₁Þ^lpþqp ðb_qÞ^lpþqp ðb₁Þ^qp1 ðb_qÞ^qp1

ðb₁Þ ðb_qÞ

1 1

det 1 a^l_i^j !

1ai;jap

:

Lemma 2.5 (cf. [Hua, 1963], Theorem 1.2.3). Let qbp>0. The following identity holds

X

l1>>lpb0

detðx^l_i^jÞ_1ai;jap

y^l₁^1þqp y_q^l1þqp ...

... y^l₁^pþqp y^l_q^pþqp y₁^qp1 y_q^qp1

...

... y1 yq

1 1

¼ Y

1ai<jap

ðxixjÞ Y

1ai<jaq

ðyiyjÞ Y^p

i¼1

Y^q

i¼1

ð1x_iy_jÞ

:

Using the above lemma, we obtain the following explicit expression for z_p;q.

Theorem 2.6. Let c0 be a constant. For a¼diagða1;. . .;apÞ and b¼ diagðb₁;. . .;b_qÞ such that a_iþb_j00, the Bessel-type function z_p;qða;bÞ is given by

z_p;qða;bÞ ¼c0

exp½trðaþbÞ Y^p

i¼1

Y^q

j¼1

ðaiþb_jÞ :

3. The Spð2n;RÞ-case

Let

Spð2n;RÞ ¼ g¼ A B B A

AMð2n;CÞ jgI_n;ng¼I_n;n

;

where AAGLðn;CÞ and BAMðn;CÞ.

A simple calculation shows that all elements A B B A

ASpð2n;RÞ satisfy AABB¼I_n; and AAB^tB¼I_n.

For a diagonal matrix a¼diagða1;. . .;a_nÞ, such that a_i00, we write z_nðaÞ ¼

ð

Spð2n;RÞ

exp½trðdiag½a;aðggÞ¹Þdg:

Remark 3.1. For n¼1 and a>0

z₁ðaÞ ¼c₀ð4aÞ¹⁼²K₁₌₂ð4aÞ,

where K_nðzÞ is the modified Bessel function of the third kind.

Let

Dn¼ fT ASymðn;CÞ jdetðInT TÞ>0g;

where Symðn;CÞ denotes the set of nn-symmetric matrices. The Spð2n;RÞ- invariant measure dmðTÞ on D_n is given by dmðtÞ ¼detðInTTÞ^ðnþ1ÞdT, where dT is the Lebesgue measure on Dn.

Using the same method used in section 2, we can deduce that if FðgÞ ¼exp½trðdiag½a;aðggÞ¹Þ; gASpð2n;RÞ;

then there exists a function F^#:Dn!C such that

F^#ðTÞ ¼exp½2 trðaÞexp½4 trðaðInTTÞ¹T TÞ:

By [Hua, 1944], every symmetric matrix ZASymðn;CÞ can be written as Z¼uLu^t, where uAUðnÞ and L¼diagðl1;. . .;lnÞ with l1bl2b blnb 0. Therefore the function F^# can be written as

F^#ðTÞ ¼exp½2 trðaÞexp½4 trðauðInL²Þ¹L²uÞ;

where L¼diagðl1;. . .;lnÞ with 1>l1bl2b blnb0 and uAUðnÞ.

As in section 2, we consider the map c:Dn!1. The image of the Lebesgue measure dT on D_n with respect to c is the measure on 1 given by

c Y

1ai<jan

ðl²_i l_j²ÞYⁿ

i¼1

lidli;

for some constant c. Thus the image of dmðTÞ ¼detðInT TÞ^ðnþ1ÞdT is

c Y

1ai<jan

ðl_i²l²_jÞYⁿ

i¼1

lið1l_i²Þ^ðnþ1Þdli:

Using the above notations and Proposition 2.2(i) for UðnÞ, we obtain z_nðaÞ ¼cexp½2 trðaÞ

ð

UðnÞ

ð

1

exp½4 trðauðInL²Þ¹L²uÞ

Y

1ai<jan

ðl²_i l_j²ÞYⁿ

i¼1

lið1l²_iÞ^ðnþ1Þdlidu

¼cexp½2 trðaÞ ð

X

ð

UðnÞ

exp½trðaudiag½x1;. . .;xnuÞdu

( )

Y

1ai<jan

ðxixjÞdx1. . .dxn:

¼ cexp½2 trðaÞ Y

1ai<jan

ðaia_jÞ ð

X

detðe^aixjÞ_1ai;jandx1. . .dxn;

where

X¼ fdiagðx1;x₂;. . .;x_nÞ jx₁bx₂b bx_nb0g:

To obtain the above second equality, we used the change of variable xi¼ 4l_i²

1l_i². Since detðe^aixjÞ_1ai;jan¼P

tASneðtÞQn

i¼1e^atðiÞxi, where Sn is the group of permutations, then

z_nðaÞ ¼ cexp½2 trðaÞ Y

1ai<jan

ðaiajÞ ð

0ax1aaxn

X

tASn

eðtÞYⁿ

i¼1

e^atðiÞxi dx_i

¼ cexp½2 trðaÞ Y

1ai<jan

ðaia_jÞ ð1

0

. . . ð1

0

X

tASn

eðtÞx₁^atð1Þ1. . .xn^{atð1ÞþþatðnÞ1}dx₁. . .dx_n

¼ cexp½2 trðaÞ Y

1ai<jan

ðaiajÞ X

tASn

eðtÞ 1

atð1Þðatð1Þþatð2ÞÞ. . .ðatð1Þþ þatðnÞÞ:

To finish the computation of z_nðaÞ, we need the following lemma.

Lemma 3.2 (cf. [Hua, 1963], Lemma 6.3.1).

X

tASN

eðtÞ 1

ltð1Þðltð1Þþltð2ÞÞ. . .ðltð1Þþ þltðNÞÞ

¼

ð1Þ^NðN1Þ=N2^N Y

1ai<jaN

ðliljÞ Y

1aiajaN

ðliþljÞ :

Using Lemma 3.2, the following theorem holds.

Theorem 3.3. Let c0 be a constant. For a¼diagða1;. . .;anÞ such that a_i00, the Bessel-type function z_nðaÞ is given by

z_nðaÞ ¼c0

exp½2 trðaÞ Y

1aiajan

ðaiþajÞ:

References

[Bre´ zin-Hikami, 2001] E. Bre´zin and S. Hikami, Characteristic polynomials of real symmetric random matrices, Commun. Math. Phys. 223 (2001), 363–382.

[Bre´ zin-Hikami, 2003] ———, An extension of the HarishChandra-Itzykson-Zuber integral, Comm. Math. Phys. 235 (2003), 125–137.

[Ben Sai¨d-Ørsted, 2003] S. Ben Saı¨d and B. Ørsted, Analysis on flat symmetric spaces, preprint (2003).

[Gross-Richards, 1989] K. I. Gross and D. P. Richards, Total positivity, spherical series, and hypergeometric functions of matrix argument, J. Appr. Theory 59 (1989), 224–246.

[Hua, 1963] L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, AMS, Providence, Rhode Island (1963).

[Hua, 1944] L. K. Hua, On the theory of automorphic functions of a matrix variable I, Amer.

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[Harish-Chandra, 1957] Harish-Chandra, Di¤erential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87–120.

[James, 1964] A. T. James, Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist. 35 (1964), 475–501.

[Macdonald, 1979] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, (1979).

[Olshanski-Vershik, 1996] G. Olshanski and A. M. Vershik, Ergodic unitarily invariant mea- sures on the space of infinite Harmitian matrices, Contemporary Mathematical Physics (R. L. Dobroshin, R. A. Minlos, M. A. Shubin, A. M. Vershik), Amer. Math. Soc.

Translations (2), 175 (1996), 137–175.

University of Aarhus

Department of Mathematical Sciences Building 530, Ny Munkegade

DK-8000 Arhus C, Denmark E-mail address: ssaid@imf.au.dk