# Note on Bessel functions of type $A_{N-1}$.

Béchir Amri

(University of Tunis, Preparatory Institut of Engineer Studies of Tunis,
Department of Mathematics, 1089 Montfleury Tunis, Tunisia
<EMAIL_ADDRESS>)

###### Abstract

Through the theory of Jack polynomials we give an iterative method for
integral formula of Bessel function of type $A_{N-1}$ and a partial product
formula for it. 111Key words and phrases: Dunkl operators, Heckman-Opdam
polynomials, Jack polynomials. 2010 Mathematics Subject Classification:
33C52,33C67, 05E05. Author partially supported by DGRST project 04/UR/15-02
and CMCU program 10G 1503.

##  1 Introduction and backgrounds

Dunkl operators which were first introduced by C. F. Dunkl [6] in the late
80ies are commuting differential-difference operators, associated to a finite
reflection groups on a Euclidean space. Their eigenfunctions are called Dunkl
kernels and appear as a generalization of the exponential functions. Although
attempts were made to study them and except the reflection group
$\mathbb{Z}_{2}^{N}$ the explicit forms or behaviors of these kernels are
remain unknown. In the present work we will be concerned with generalized
Bessel functions $J_{k}$ defined through symmetrization of Dunkl kernels in
the case of the symmetric group $S_{N}$. We will obtain the following

$\displaystyle
J_{k}(\mu,\lambda)=\int_{\mathbb{R}^{N-1}}e^{\langle\mu,x\rangle}\delta_{k}(\lambda,x)dx.$
(1.1)

where the function $\delta_{k}$ can be explicitly computed using a recursive
formula on the dimension $N$. The key ingredient is the integral formula of A.
Okounkov and G. Olshanski [10] for Jack polynomials. As the last are connected
with Heckman-opdam-Jacobi polynomials [2] the formula (1.1) follow by limit
transition. We should note here that when $N=3$, the formula (1.1) is
comparable to that obtained by C. F. Dunkl [5] for intertwining operator.

Let us start with some well-known facts about Heckman Opdam Jacobi
polynomials, Jack polynomials and Dunkl kernels associated with a root system
$R$. The standard references are [2, 4, 8, 11, 16, 15]. Here $\mathbb{R}^{N}$
is equipped with the usual inner product $\langle,\;.\;,\rangle$ and the
canonical orthonormal basis $(e_{1},e_{2},...,e_{N})$. Further, we shall
assume that $R$ is reduced and crystallographic, that is a finite subset of
$\mathbb{R}^{N}\backslash\\{0\\}$ which satisfies:
(i) $R$ spanned $\mathbb{R}^{N}$.
(ii) $R$ is invariant under $r_{\alpha}$ the reflection in the hyperplane
orthogonal to any $\alpha\in R$.
(iii) $\alpha.\mathbb{R}\cap R=\\{\pm\alpha\\}$ for all $\alpha\in R$
(iii) for all $\alpha,\;\beta\in R$;
$\langle\alpha,\breve{\beta}\rangle\in\mathbb{Z}$,
$\breve{\beta}=\frac{2\beta}{\|\beta\|^{2}}$
We assume that the reader is familiar with the basics of root systems and
their Weyl groups, see for examples Humphreys [9].

###  1.a  Heckman Opdam Jacobi polynomials.

Let $R$ be a reduced root system with $\\{\alpha_{1},...,\alpha_{N}\\}$ be a
basis of simple roots and $R_{+}$ be the set of positive roots determined by
this basis. The fundamental weights $\\{\beta_{1},...,\beta_{N}\\}$ are given
by $\langle\;\beta_{j},\;\check{\alpha}_{i}\;\rangle=\delta_{i,j}$,
$\displaystyle{\check{\alpha_{i}}=\frac{2\alpha_{i}}{\|\alpha_{i}\|^{2}}}$.
Let $\displaystyle Q=\bigoplus_{i=1}^{N}\mathbb{Z}\alpha_{i}$, $\displaystyle
P=\bigoplus_{i=1}^{N}\mathbb{Z}\beta_{i}$, $\displaystyle
Q^{+}=\bigoplus_{i=1}^{N}\mathbb{N}\alpha_{i}$ and $\displaystyle
P^{+}=\bigoplus_{i=1}^{N}\mathbb{N}\beta_{i}$. We define a partial ordering on
$P$ by $\lambda\preceq\mu$ if $\mu-\lambda\in Q^{+}$

The group algebra $\mathbb{C}[P]$ of the free Abelian group $P$ is the algebra
generated by the formal exponentials $e^{\lambda}$, $\lambda\in P$ subject to
the multiplication relation $e^{\lambda}e^{\mu}=e^{\lambda+\mu}$. The Weyl
group W acts on $\mathbb{C}[P]$ by $we^{\lambda}=e^{w\lambda}$. The orbit-sums
$\displaystyle m_{\lambda}=\sum_{\mu\in W.\lambda}e^{\mu}$, $\lambda\in P^{+}$
form a basis of $\mathbb{C}[P]^{W}$, the subalgebra of $W$-invariant elements
of $\mathbb{C}[P]$. Here $W.\lambda$ denotes the W-orbit of $\lambda$.

Let $\mathbb{T}=\mathbb{R}^{d}/2\pi\check{Q}$ where
$\displaystyle\check{Q}=\bigoplus_{i=1}^{d}\mathbb{Z}\check{\alpha_{i}}$. The
algebra $\mathbb{C}[P]$ can be realized explicitly as the algebra of
polynomials on the torus $\mathbb{T}$ through the identification
$e^{\lambda}(\dot{x})=e^{i\langle\lambda,x\rangle}$ where
$\dot{x}\in\mathbb{T}$ is the image of $x\in\mathbb{R}^{d}$. Let
$k:R\rightarrow[0,+\infty[$ be a $W$-invariant function, called multiplicity
function. We equip $C[P]^{W}$ with the inner product

$(f,g)_{k}=\int_{\mathbb{T}}f(x)\overline{g(x)}\delta_{k}(x)dx$

where

$\delta_{k}=\prod_{\alpha\in
R^{+}}\left|e^{\frac{\alpha}{2}}-e^{-\frac{\alpha}{2}}\right|^{2k_{\alpha}}$

and $dx$ is the Haar measure on $\mathbb{T}$.

The Heckman Opdam Jacobi polynomials are introduced by Heckman and Opdam [8]
as the unique family of elements $P_{\lambda}\in\mathbb{C}[P]^{W}$,
$\lambda\in P^{+}$ satisfying the following conditions:

  * (i)

$P_{\lambda}=m_{\lambda}+\sum_{\mu\prec\lambda}a_{\lambda\mu}m_{\mu}$

  * (ii)

$\langle P_{\lambda},m_{\mu}\rangle=0$ if $\mu\in P_{+}$, $\lambda\prec\mu$.

( Note that in [8], these polynomials are indexed by $-P_{+}$ instead of
$P_{+}$ ). They form an orthogonal basis of $\mathbb{C}[P]^{W}$ and satisfy
the second differential equation

$\Big{(}\Delta+\sum_{\alpha\in R_{+}}k_{\alpha}\coth(\frac{1}{2}\langle
x,\alpha\rangle)\partial_{\alpha}\Big{)}P_{\lambda}(x)=\langle\lambda,\lambda+\sum_{\alpha\in
R_{+}}k_{\alpha}\alpha\rangle P_{\lambda}(x).$

where $\Delta$ is the Laplace operator on $\mathbb{R}^{N}$.

The Cherednik operator $T_{\xi}$, $\xi\in\mathbb{R}^{N}$, associated with the
root system $R$ and the multiplicity $k$ is defined by

$T_{\xi}^{k}=\partial_{\xi}+\sum_{\alpha\in
R_{+}}k_{\alpha}\langle\alpha,\;\xi\rangle\;\frac{1-r_{\alpha}}{1-e^{{}^{\alpha}}}-\langle\rho_{k},\;\xi\rangle,$

where $\displaystyle{\rho_{k}=\frac{1}{2}\sum_{\alpha\in
R_{+}}k_{\alpha}\alpha}$. The hypergeometric function $F_{k}$ is defined as
the unique holomorphic W-invariant function on
$\mathbb{C}^{N}\times(\mathbb{R}^{N}+iU)$ ( U is a W-invariant neighborhood of
$0$ ) which satisfies the system of differential equations:

$p(T_{e_{1}},...T_{e_{N}})F_{k}(\lambda,.)=p(\lambda)F_{k}(\lambda,.);\qquad
F(\lambda,0)=1$

for all $\lambda\in\mathbb{C}^{N}$ and all $W$-invariant polynomial $p$ on
$\mathbb{R}^{N}$. The Heckman opdam Jacobi polynomials are related to the
hypergeometric function $F_{k}$ by ( see [7] )

$\displaystyle
F_{k}(\lambda+\rho_{k},x)=c(\lambda+\rho_{k})P_{\lambda}(x);\qquad\lambda\in
P^{+},\;x\in\mathbb{R}^{N},$ (1.2)

where the function $c$ is given on $\mathbb{R}^{N}$ by

$\displaystyle c(\lambda)=\prod_{\alpha\in
R^{+}}\frac{\Gamma(\langle\lambda,\check{\alpha}\rangle)\Gamma(\langle\rho,\check{\alpha}\rangle+k_{\alpha})}{\Gamma(\langle\lambda,\check{\alpha}\rangle+k_{\alpha})\Gamma(\langle\rho,\check{\alpha}\rangle)}.$
(1.3)

###  1.b  Jack polynomials

Let $k>0$, the symmetric group $S_{N}$ acts on the ring of polynomials
$\mathbb{Q}(k)[x_{1},...,x_{N}]$ by

$\tau p(x_{1},...,x_{N})=p(x_{\tau(1)},...,x_{\tau(N)})$

Let $\Lambda_{N}$ the subspace of symmetric polynomials,

$\Lambda_{N}=\\{p\in\mathbb{C}[x_{1},...,x_{N}],\;\tau p=p,\;\forall\tau\in
S_{N}\\}.$

We call partition all $\lambda=(\lambda_{1},...\lambda_{N})\in\mathbb{N}^{N}$
such that $\lambda_{1}\geq...\geq\lambda_{N}$. The weight of a partition
$\lambda$ is the sum $|\lambda|=\lambda_{1}+...+\lambda_{N}$ and its length
$\ell(\lambda)=\max\left\\{j;\;\lambda_{j}\neq 0\right\\}.$ The set of all
partitions are partially ordered by the dominance order:

$\lambda\leq\mu\Leftrightarrow|\lambda|=|\mu|\quad\text{
and}\quad\lambda_{1}+\lambda_{2}+...+\lambda_{i}\leq\mu_{1}+\mu_{2}+...+\mu_{i}$

for all $i=1,2,...,N$. The simplest basis of $\Lambda_{N}$ is given by the
monomial symmetric polynomials,

$m_{\lambda}(x)=\sum_{\mu\in S_{N}\lambda}x_{1}^{\mu_{1}}...x_{N}^{\mu_{N}}.$

We define an inner product on $\Lambda_{N}$ by

$\langle
f,g\rangle_{k}=\int_{T}f(z)\overline{g(z)}\prod_{i<j}|z_{i}-z_{j}|^{2k}dz$

where $T=\\{(z_{1},...,z_{N})\in\mathbb{C}^{N};|z_{j}|=1,\;\forall\;1\leq
j\leq N\\}$ is the $N$-dimensional torus and $dz$ is the haar measure on $T$.
Jack symmetric polynomials $j_{\lambda}$ indexed by a partitions $\lambda$ can
be defined as the unique polynomials such that

  * (i)

$j_{\lambda}=m_{\lambda}+\sum_{\mu\prec\lambda}m_{\mu}$,

  * (ii)

$\langle j_{\lambda},m_{\mu}\rangle_{k}=0$ if $\lambda\leq\mu$.

By a result of I. G. Macdonald ([13], p: 383 ) they form a family of
orthogonal polynomials. Jack polynomials can be defined as eigenfunctions of
certain Laplac-Beltrami type operator ( coming in the theory of Calogero
integrable systems and in random matrix theory ),

$L_{k}=\sum_{i=1}^{d}x_{i}^{2}\frac{\partial^{2}}{\partial
x_{i}^{2}}+2k\sum_{i\neq
j}\frac{x_{i}^{2}}{x_{i}-x_{j}}\frac{\partial}{\partial x_{i}}.$

Jack polynomials $j_{\lambda}$ are homogeneous of degree $|\lambda|$ and
satisfy the compatibility relation

$\displaystyle
j_{(\lambda_{1},...,\lambda_{N-1},0)}(x_{1},...,x_{N-1},0)=j_{(\lambda_{1},...,\lambda_{N-1})}(x_{1},...,x_{N-1}).$
(1.4)

The relationship between Heckman Opdam Jacobi polynomials and Jack polynomials
can be illustrated as follows ( see [2] ): Let $\mathbb{V}$ be the hyperplane
orthogonal to the vector $e=e_{1}+...+e_{N}$. In $V$ we consider the root
system of type $A_{N-1}$,

$R_{A}=\\{\pm(e_{i}-e_{j}),\;1\leq i<j\leq N\\}.$

The fundamental weights are given by $\pi_{N}(\omega_{i})$,
$\omega_{i}=e_{1}+...+e_{i}$, where $\pi_{N}$ denote the orthogonal projection
along $e$ onto V,

$\pi_{N}(x)=x-\frac{1}{N}\left(\sum_{i=1}^{N}x_{i}\right)e=\left(x_{1}-\frac{1}{N}\left(\sum_{i=1}^{N}x_{i}\right),...,x_{N}-\frac{1}{N}\left(\sum_{i=1}^{N}x_{i}\right)\right)$

and then $P_{A}^{+}=\\{\pi_{N}(\lambda),\lambda\;\text{partition}\\}$. The
result is that:

$\displaystyle j_{\lambda}(e^{x})=P_{\pi_{N}(\lambda)}(x),$ (1.5)

For all partition $\lambda$ and all $x\in\mathbb{V}$ with
$e^{x}=(e^{x_{1}},...,e^{x_{N}})$.

###  1.c Dunkl kernels and Dunkl-Bessel functions

The Dunkl operator $D_{\xi}$, $\xi\in\mathbb{R}^{N}$ associated with a root
system $R$ and a multiplicity function $k$ is defined by

$D_{\xi}=\partial_{\xi}+\sum_{\alpha\in
R^{+}}k(\alpha)\langle\alpha,\xi\rangle\frac{1-r_{\alpha}}{\langle\alpha,.\rangle}.$

The Dunkl intertwining operator $V_{k}$ is the unique isomorphism on the
polynomials space $\mathbb{C}[\mathbb{R}^{N}]$ such that

$V_{k}(1)=1,\quad V_{k}(\mathcal{P}_{n})=\mathcal{P}_{n}\quad\text{and}\quad
D_{\xi}V_{k}=V_{k}\partial_{\xi}$

where $\mathcal{P}_{n}$ is the subspace of homogeneous polynomials of degree
$n\in\mathbb{N}$. For $r>0$ , $V_{k}$ extends to a continuous linear operator
on the Banach space

$A_{r}=\\{f=\sum_{n=0}^{\infty}f_{n},\;f_{n}\in\mathcal{P}_{n},\;\|f\|_{A_{r}}=\sum_{n=0}^{\infty}\sup_{|x|\leq
r}|f_{n}(x)|<\infty\\}$

by

$V_{k}(f)=\sum_{n=0}^{\infty}V_{k}(f_{n}).$

A remarkable result due to M. Rösler [14] says that for each
$x\in\mathbb{R}^{N}$,

$V_{k}(f)(x)=\int_{\mathbb{R}^{d}}f(\xi)d\mu_{x}(\xi)$

where $\mu_{x}$ is a probability measure supported in $co(x)$ the convex hull
of the orbit W.x.

The Dunkl kernel $E_{k}$ is given by

$\displaystyle
E_{k}(x,y)=V_{k}(e^{\langle\;.\;,\;y\;\rangle})(x)=\int_{\mathbb{R}^{d}}e^{\langle\xi,y\rangle}d\mu_{x}(\xi),\quad
x\in\mathbb{R}^{N},\;y\in\mathbb{C}^{N}$

and having the following properties:

  * (i)

For each $y\in\mathbb{C}^{N}$ the function $E_{k}(.,y)$ is the unique solution
of eigenvalue problem:

$D_{\xi}f(x)=\langle\xi,y\rangle
f(x)\;\forall\;\quad\xi\in\mathbb{R}^{N}\;\text{and}\;f(0)=1.$

  * (ii)

$E_{k}$ extends to a holomorphic function on
$\mathbb{C}^{N}\times\mathbb{C}^{N}$ and for all
$(x,y)\in\mathbb{C}^{N}\times\mathbb{C}^{N}$, $w\in W$ and $t\in\mathbb{C}$:

$E_{k}(x,y)=E_{k}(y,x),\quad E_{k}(wx,wy)=E_{k}(x,y)\quad\text{and}\quad
E_{k}(x,ty)=E_{k}(tx,y)$

We define the Bessel function associated with $R$ and $k$ by,

$\displaystyle J_{k}(x,y)=\frac{1}{|W|}\sum_{w\in W}E_{k}(x,wy).$

The limit transition between hypergeometric functions $F_{k}$ and Dunkl Bessel
function is expressed by ( see (2.21) of [14] )

$\displaystyle
J_{k}(x,y)=\lim_{n\rightarrow+\infty}F_{k}(nx+\rho_{k},\frac{y}{n})\;.$ (1.6)

According to these preliminaries we can now formulate the main result of this
note.

##  2 Integral formula for $J_{k}$

The starting point is the following remarkable integral identity obtained by
[10] which connecting jack polynomials of $N$ variables to Jack polynomials of
$N-1$ variables. For $\lambda=(\lambda_{1},...,\lambda_{N})\in\mathbb{R}^{N}$
we use the notation $|\lambda|=\lambda_{1}+...+\lambda_{N}$.

######  Proposition 1 ([10]).

Suppose that the partition $\mu$ has less than $N$ parts and
$\lambda\in\mathbb{R}^{N}$ such that $\lambda_{1}\geq...\geq\lambda_{N}$. Then

$\displaystyle
j_{\mu}(\lambda)=\frac{1}{U(\mu)V(\lambda)^{2k-1}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}j_{\mu}(\nu)V(\nu)\Pi(\lambda,\nu)d\nu$
(2.1)

where

$U(\mu)=\prod_{j=1}^{N-1}\beta(\mu_{j}+(N-j)k,k),\quad V(\lambda)=\prod_{1\leq
i<j\leq N}(\lambda_{i}-\lambda_{j})$

and

$\Pi(\lambda,\nu)=\prod_{i\leq
j}(\lambda_{i}-\nu_{j})^{k-1}\prod_{i>j}(\nu_{j}-\lambda_{i})^{k-1}.$

We follow three simple steps that lead to our formula. All functions of $N$
variables will be indexed by $N$ and by $N-1$ if it considered as $N-1$
variables.
Step 1: For any partition $\mu=(\mu_{1},...,\mu_{N})$ we set

$\widetilde{\mu}=(\mu_{1}-\mu_{N},...,\mu_{N-1}-\mu_{N},0)\quad\text{and}\quad\overline{\mu}=(\mu_{1}-\mu_{N},...,\mu_{N-1}-\mu_{N})\in\mathbb{R}^{N-1}.$

By Homogeneity of Jack polynomials we have that

$j_{\mu,N}(\lambda)=\left(\prod_{j=1}^{N}\lambda_{j}\right)^{\mu_{N}}j_{\widetilde{\mu},N}(\lambda)$

and from (2.1) and (1.4) we may write

$j_{\mu,N}(\lambda)=\frac{\left(\prod_{j=1}^{N}\lambda_{j}\right)^{\mu_{N}}}{U_{N}(\widetilde{\mu})V_{N}(\lambda)^{2k-1}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}j_{\overline{\mu},N-1}(\nu)V(\nu)\Pi(\lambda,\nu)d\nu.$

Taking $\lambda$ in $\mathbb{V}$ and making use a change of variables we get
that

$\displaystyle j_{\mu,N}(e^{\lambda})$ $\displaystyle=$
$\displaystyle\frac{1}{U_{N}(\widetilde{\mu})V_{N}(e^{\lambda})^{2k-1}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}e^{|\nu|}j_{\overline{\mu},N-1}(e^{\nu})V_{N-1}(e^{\nu})\Pi_{N}(e^{\lambda},e^{\nu})d\nu$
$\displaystyle=$
$\displaystyle\frac{1}{U_{N}(\widetilde{\mu})V_{N}(e^{\lambda})^{2k-1}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}e^{|\nu|(1+\frac{|\overline{\mu}|}{N-1})}j_{\overline{\mu},N-1}(e^{\pi_{N-1}(\nu)})V_{N-1}(e^{\nu})\Pi_{N}(e^{\lambda},e^{\nu})d\nu.$

In order by (1.2), (1.5) and (1.3) we have

$F_{N}(\pi_{N}(\mu)+\rho_{k,N},\lambda)=c_{N}(\pi_{N}(\mu)+\rho_{k,N})j_{\mu,N}(e^{\lambda})=c_{N}(\mu+\rho_{k,N})j_{\mu,N}(e^{\lambda}),$

where here

$\rho_{k,N}=\frac{k}{2}\sum_{i=1}^{N}(N-2i+1)e_{i}=\left(\frac{k(N-1)}{2},...,\frac{k(N-2i+1)}{2},...,\frac{-k(N-1)}{2}\right)\in\mathbb{R}^{N}$

and

$c_{N}(\mu+\rho_{k,N})=\prod_{1\leq i<j\leq
N}\frac{\Gamma(\mu_{i}-\mu_{j})\Gamma(k(j-i+1))}{\Gamma(\mu_{i}-\mu_{j}+k)\Gamma(k(j-i))}.$

Therefore,

$\displaystyle
F_{N}(\pi_{N}(\mu)+\rho_{k,N},\lambda)=\frac{c_{N}(\mu+\rho_{k,N})}{c_{N-1}(\overline{\mu}+\rho_{k,N-1})U_{N}(\widetilde{\mu})V_{N}(e^{\lambda})^{2k-1}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}e^{|\nu|(1+\frac{|\overline{\mu}|}{N-1})}F_{N-1}(\pi_{N-1}(\overline{\mu})+\rho_{k,N-1},\pi_{N-1}(\nu))V_{N-1}(e^{\nu})\Pi_{N}(e^{\lambda},e^{\nu})d\nu.$

Step 2: Now we apply (1.6), by using the following when $n\rightarrow+\infty$

$\displaystyle U_{N}(n\widetilde{\mu})\sim
n^{-k(N-1)}\Gamma(k)^{N-1}\prod_{j=1}^{N-1}(\mu_{j}-\mu_{N})^{-k},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle V_{N}(e^{\frac{\lambda}{n}})\sim
n^{-\frac{N(N-1)}{2}}V_{N}(\lambda),$ $\displaystyle
c_{N}(n\mu+\rho_{k,N})\sim n^{\frac{-kN(N-1)}{2}}V_{N}(\mu)^{-k}\prod_{1\leq
i<j\leq N}\frac{\Gamma(k(j-i+1))}{\Gamma(k(j-i))},$ $\displaystyle
c_{N-1}(n\overline{\mu}+\rho_{k,N-1})\sim
n^{\frac{-k(N-1)(N-2)}{2}}V_{N-1}(\overline{\mu})^{-k}\prod_{1\leq i<j\leq
N-1}\frac{\Gamma(k(j-i+1))}{\Gamma(k(j-i))},$
$\displaystyle\frac{c_{N}(n\mu+\rho_{k,N})}{c_{N-1}(n\overline{\mu}+\rho_{k,N-1})}\sim
n^{-k(N-1)}\frac{\Gamma(Nk)}{\Gamma(k)}\prod_{j=1}^{N-1}(\mu_{j}-\mu_{N})^{-k},$
$\displaystyle\Pi_{N}(e^{\frac{\lambda}{n}},e^{\frac{\nu}{n}})\sim
n^{-N(N-1)(k-1)}\Pi(\lambda,\nu).$

Thus

$\displaystyle J_{k,N}(\pi_{N}(\mu),\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}$
(2.2) $\displaystyle\qquad\qquad
e^{|\overline{\mu}|\frac{|\nu|}{N-1}}J_{k,N-1}(\pi_{N-1}(\overline{\mu}),\pi_{N-1}(\nu))V(\nu)\Pi(\lambda,\nu)d\nu.$

Step 3:
The formula (2.2) is valid only for a partition $\mu$, to keep it for any
$\mu\in\mathbb{R}^{N}$ we proceed as follows. Let $r\in(0,+\infty)$ and $\mu$
be a partition. We obtain after a change of variables

$\displaystyle J_{k,N}(\pi_{N}(r\mu),\lambda)=J_{k,N}(\pi_{N}(\mu),r\lambda)$
$\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}e^{|\overline{r\mu}|\frac{|\nu|}{N-1}}$
$\displaystyle\qquad\qquad
J_{k,N-1}(\pi_{N-1}(\overline{r\mu}),\pi_{N-1}(\nu))V(\nu)\Pi(\lambda,\nu)d\nu.$

Since the set $\\{r\mu;\quad r\in(0,+\infty),\quad\mu\;\text{partitions}\;\\}$
is dense in the set

$H=\\{\mu\in\mathbb{R}^{N},\quad 0\leq\mu_{N}\leq...\leq\mu_{1}\\}$

and $J_{k,N}$ is $S_{N}$-invariant continuous function then (2.2) can be
extended to all $\mu\in H$. Now for $\mu\in\mathbb{R}^{N}$ we denote by
$\mu^{+}$ the unique element of $S_{N}.\mu$ so that
$\mu^{+}_{N}\leq...\leq\mu^{+}_{1}$. So we have
$J_{k,N}(\pi_{N}(\mu),\lambda)=J_{k,N}(\pi_{N}(\mu^{+}),\lambda)=J_{k,N}(\pi_{N}(\widetilde{\mu^{+}}),\lambda)$
and since $\widetilde{\mu^{+}}\in H$ then

$\displaystyle J_{k,N}(\pi_{N}(\mu),\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\lambda_{2}}^{\lambda_{1}}...\int_{\lambda_{N}}^{\lambda_{N-1}}$
$\displaystyle\qquad\qquad
e^{|\overline{\mu^{+}}|\frac{|\nu|}{N-1}}J_{k,N-1}(\pi_{N-1}(\overline{\mu^{+}}),\pi_{N-1}(\nu))V(\nu)\Pi(\lambda,\nu)d\nu.$

Now when restricted to the space $\mathbb{V}$ we state the following.

######  Theorem 1.

For all $\mu,\lambda\in\mathbb{V}$ we have

$\displaystyle J_{k,N}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\lambda_{2}^{+}}^{\lambda_{1}^{+}}...\int_{\lambda_{N}^{+}}^{\lambda_{N-1}^{+}}e^{|\overline{\mu^{+}}|\frac{|\nu|}{N-1}}$
(2.3) $\displaystyle\qquad\qquad
J_{k,N-1}(\pi_{N-1}(\overline{\mu^{+}}),\pi_{N-1}(\nu))V_{N-1}(\nu)\Pi_{N}(\lambda^{+},\nu)d\nu.$

In what follows, we shall restrict ourselves to the case $N=2,3,4$ where we
give representations of $J_{k,N}$ as Laplace-type integrals,

$\displaystyle
J_{k,N}(\mu,\lambda)=\int_{\mathbb{R}^{N}}e^{\langle\mu,x\rangle}d\nu_{\lambda}(x).$

where $\nu_{\lambda}$ is a probability measure supported in the convex hall of
the orbit $S_{N}.\lambda$.

###  2.a Bessel function of type $A_{1}$

When $N=2$ we have that $\mathbb{V}=\mathbb{R}(e_{1}-e_{2})$,
$\mu^{+}=(|\mu_{1}|,-|\mu_{1}|)$, $\overline{\mu^{+}}=2|\mu_{1}|$ and
$\lambda^{+}=(|\lambda_{1}|,-|\lambda_{1}|)$. It is obvious that $J_{k,1}=1$,
so we get from (2.3)

$\displaystyle J_{k,2}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(2k)}{(\Gamma(k))^{2}(2|\lambda_{1}|)^{2k-1}}\int_{-|\lambda_{1}|}^{|\lambda_{1}|}e^{2|\mu_{1}|\nu}(\lambda_{1}^{2}-\nu^{2})^{k-1}d\nu$
$\displaystyle=$
$\displaystyle\frac{\Gamma(k+\frac{1}{2})}{\sqrt{\pi}\Gamma(k)}\int_{-1}^{1}e^{(2|\mu_{1}||\lambda_{1}|\nu}(1-\nu^{2})^{k-1}d\nu$
$\displaystyle=$
$\displaystyle\mathcal{J}_{k-\frac{1}{2}}(2\mu_{1}\lambda_{1})$

where $\mathcal{J}_{k-\frac{1}{2}}$ is the modified Bessel function given by

$\mathcal{J}_{k-\frac{1}{2}}(z)=\Gamma(k+\frac{1}{2})\sum_{n=0}^{\infty}\frac{1}{n!\Gamma(n+k+\frac{1}{2})}(\frac{z}{2})^{2n}.$

However, it is usual to identify $\mathbb{V}=\mathbb{R}\varepsilon$,
$\varepsilon=\frac{e_{1}-e_{2}}{\sqrt{2}}$ with $\mathbb{R}$ and write

$\displaystyle
J_{k,2}(\mu,\lambda)=\mathcal{J}_{k-\frac{1}{2}}(\mu\lambda),\qquad\mu,\lambda\in\mathbb{R}.$

###  2.b Bessel function of type $A_{2}$

Let $\mu=(\mu_{1},\mu_{2},\mu_{3})$ and
$\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})$ in the fundamental Weyl
chamber

$C=\\{(u_{1},u_{2},u_{3});\quad u_{1}\geq u_{2}\geq u_{3},\quad
u_{1}+u_{2}+u_{3}=0\\}.$

With $\overline{\mu}=(\mu_{1}-\mu_{3},\mu_{2}-\mu_{3},)$ and
$\pi_{2}(\overline{\mu})=(\frac{\mu_{1}-\mu_{2}}{2},\frac{\mu_{2}-\mu_{1}}{2})$
the formula (2.3) gives

$\displaystyle J_{k,3}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(3k)}{V(\lambda)^{2k-1}\Gamma(k)^{3}}\int_{\lambda_{2}}^{\lambda_{1}}\int_{\lambda_{3}}^{\lambda_{2}}e^{\frac{(\mu_{1}+\mu_{2}-2\mu_{3})(\nu_{1}+\nu_{2})}{2}}\mathcal{J}_{k-\frac{1}{2}}(\frac{(\mu_{1}-\mu_{2})(\nu_{1}-\nu_{2})}{2})(\nu_{1}-\nu_{2})$
$\displaystyle\qquad\Big{(}(\lambda_{1}-\nu_{1})(\lambda_{1}-\nu_{2})(\lambda_{2}-\nu_{2})(\nu_{1}-\lambda_{2})(\nu_{1}-\lambda_{3})(\nu_{2}-\lambda_{3})\Big{)}^{k-1}d\nu_{1}d\nu_{2}.$

Using the change of variables: $x=\frac{\nu_{1}+\nu_{2}}{2}$,
$z=\frac{\nu_{1}-\nu_{2}}{2}$ we have

$\displaystyle J_{k,3}(\mu,\lambda)=$
$\displaystyle\frac{4\Gamma(3k)}{V(\lambda)^{2k-1}\Gamma(k)^{3}}\int_{\mathbb{R}}\int_{\mathbb{R}}ze^{(\mu_{1}+\mu_{2}-2\mu_{3})x}\mathcal{J}_{k-\frac{1}{2}}(\mu_{1}-\mu_{2})z)\;\chi_{[\lambda_{2},\lambda_{1}]}(x+z)\;\chi_{[\lambda_{3},\lambda_{2}]}(x-z)$
(2.4)
$\displaystyle\Big{(}(\lambda_{1}-x)^{2}-z^{2})((\lambda_{3}-x)^{2}-z^{2})(z^{2}-(\lambda_{2}-x)^{2})\Big{)}^{k-1}dxdz.$

Now recall that

$\displaystyle\mathcal{J}_{k-\frac{1}{2}}((\mu_{1}-\mu_{2})z)$
$\displaystyle=$
$\displaystyle\frac{\Gamma(2k)}{2^{2k-1}\Gamma(k)^{2}}\int_{\mathbb{R}}e^{(\mu_{1}-\mu_{2})zt}(1-t^{2})^{k-1}\chi_{[-1,1]}(t)dt.$
(2.5) $\displaystyle=$
$\displaystyle\frac{\Gamma(2k)}{2^{2k-1}\Gamma(k)^{2}}\int_{\mathbb{R}}e^{(\mu_{1}-\mu_{2})y}(1-\frac{y^{2}}{z^{2}})^{k-1}\chi_{[-1,1]}(\frac{y}{z})z^{-1}dy$

then inserting (2.5) in (2.4) with the use of Fubini’s Theorem we can write

$\displaystyle
J_{k,3}(\mu,\lambda)=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{(\mu_{1}+\mu_{2}-2\mu_{3})x+(\mu_{1}-\mu_{2})y}\Delta_{k}(\lambda,x,y)dxdy$

where

$\displaystyle\Delta_{k}(\lambda,x,y)=$
$\displaystyle\frac{4\Gamma(2k)\Gamma(3k)}{2^{2k-3}\Gamma(k)^{5}V(\lambda)^{2k-1}}\int_{\mathbb{R}}\left(\frac{z^{2}-y^{2}}{z^{2}}\right)^{k-1}\Big{(}(\lambda_{1}-x)^{2}-z^{2})((\lambda_{3}-x)^{2}-z^{2})(z^{2}-(\lambda_{2}-x)^{2})\Big{)}^{k-1}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\chi_{[-1,1]}(\frac{y}{z})\chi_{[\lambda_{1},\lambda_{2}]}(x+z)\chi_{[\lambda_{3},\lambda_{2}]}(x-z)dz$

We note here that

$\displaystyle\chi_{[-1,1]}(\frac{y}{z})\chi_{[\lambda_{1},\lambda_{2}]}(x+z)\chi_{[\lambda_{3},\lambda_{2}]}(x-z)=\chi_{\max(|y|,|x-\lambda_{2}|)\leq
z\leq\min(x-\lambda_{3},\lambda_{1}-x)}.$

Thus we have

$\displaystyle\Delta_{k}(\lambda,x,y)=$
$\displaystyle\frac{4\Gamma(2k)\Gamma(3k)}{2^{2k-3}\Gamma(k)^{5}V(\lambda)^{2k-1}}\int_{\max(|y|,|x-\lambda_{2}|)}^{\min(x-\lambda_{3},\lambda_{1}-x)}\left(\frac{z^{2}-y^{2}}{z^{2}}\right)^{k-1}\qquad\qquad$
$\displaystyle\qquad\Big{(}(\lambda_{1}-x)^{2}-z^{2})((\lambda_{3}-x)^{2}-z^{2})(z^{2}-(\lambda_{2}-x)^{2})\Big{)}^{k-1}dz$

if

$\displaystyle\max(|y|,|x-\lambda_{2}|)\leq\min(x-\lambda_{3},\lambda_{1}-x)$

and $\Delta_{k}(\lambda,x,y)=0$, otherwise. Making the change of variables

$\nu_{1}=x+y,\qquad\nu_{2}=x-y$

and put $\nu=(\nu_{1},\nu_{2},\nu_{3})\in\mathbb{V}$ with
$\nu_{3}=-(\nu_{1}+\nu_{2})$ we obtain

$\displaystyle
J_{3}^{k}(\mu,\lambda)=\frac{1}{2}\int_{\mathbb{R}^{2}}e^{\mu_{1}\nu_{1}+\mu_{2}\nu_{2}+\mu_{3}\nu_{3}}\Delta_{k,2}\left(\lambda,\frac{\nu_{1}+\nu_{2}}{2},\frac{\nu_{1}-\nu_{2}}{2}\right)d\nu_{1}d\nu_{2}$

But we can identify $\mathbb{R}^{2}$ with the space $\mathbb{V}$ via the basis
$(e_{1}-e_{2},e_{2}-e_{3})$, since for
$\nu=(\nu_{1},\nu_{2},\nu_{3})\in\mathbb{V}$ we have
$\nu=\nu_{1}(e_{1}-e_{2})+\nu_{2}(e_{1}-e_{3})$. Then we get

$\displaystyle
J_{k,3}(\mu,\lambda)=\int_{\mathbb{R}^{2}}e^{\langle\mu,\nu\rangle}\delta_{k,2}\left(\lambda,\nu\right)d\nu_{1}d\nu_{2}.$
(2.6)

with

$\displaystyle\delta_{k,2}(\lambda,\nu)=\frac{1}{2}\Delta_{k,2}\left(\lambda,\frac{\nu_{1}+\nu_{2}}{2},\frac{\nu_{1}-\nu_{2}}{2}\right).$

Now considering the orthonormal basis $(\varepsilon_{1},\varepsilon_{2})$ of
$\mathbb{V}$,

$\varepsilon_{1}=\frac{1}{\sqrt{6}}(e_{1}+e_{2}-2e_{3}),\quad\varepsilon_{2}=\frac{1}{\sqrt{2}}(e_{1}-e_{2})$

we can write

$\mu=\frac{(\mu_{1}+\mu_{2}-2\mu_{3})}{\sqrt{6}}\;\varepsilon_{1}+\frac{\mu_{1}-\mu_{2}}{\sqrt{2}}\;\varepsilon_{2}$

and for $x=x_{1}\varepsilon_{1}+x_{2}\varepsilon_{2}$

$\langle\mu,x\rangle=\frac{(\mu_{1}+\mu_{2}-2\mu_{3})}{\sqrt{6}}x_{1}+\frac{\mu_{1}-\mu_{2}}{\sqrt{2}}x_{2}$

Then using change of variables $x_{1}=\sqrt{6}\;x$ and $x_{2}=\sqrt{2}\;y$ in
the formula (LABEL:k3) we obtain

$\displaystyle
J_{3}^{k}(\mu,\lambda)=\frac{1}{\sqrt{12}}\int_{\mathbb{R}^{2}}e^{\langle\mu,x\rangle}\Delta_{k}\left(\lambda,\frac{x_{1}}{\sqrt{6}},\frac{x_{2}}{\sqrt{2}}\right)dx_{1}dx_{2}.$

######  Proposition 2.

For all $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})\in C$ the function
$\delta_{k,2}\left(\lambda,.\right)$ is supported in the closed convex hull
$co(\lambda)$ of the $S_{3}$-orbit of $\lambda$, described by:
$\nu=(\nu_{1},\nu_{2},\nu_{3})\in\mathbb{V}$ such that

$\displaystyle\lambda_{3}\leq\min(\nu_{1},\nu_{2},\nu_{3})\leq\max(\nu_{1},\nu_{2},\nu_{3})\leq\lambda_{1}.$

###### Proof.

In view of (2.b) and (2.b) the support of $\delta_{k,2}\left(\lambda,.\right)$
is contain is the set

$\left\\{\nu\in\mathbb{V};\quad\max\left(\frac{|\nu_{1}-\nu_{2}|}{2},\left|\frac{\nu_{1}+\nu_{2}}{2}-\lambda_{2}\right|\right)\leq\min\Big{(}\frac{\nu_{1}+\nu_{2}}{2}-\lambda_{3},\lambda_{1}-\frac{\nu_{1}+\nu_{2}}{2}\Big{)}\right\\}$

which by straightforward calculus reduced to the set

$\\{\nu\in\mathbb{V};\quad\lambda_{3}\leq\min(\nu_{1},\nu_{2},\nu_{3})\leq\max(\nu_{1},\nu_{2},\nu_{3})\leq\lambda_{1}\\}.$

However, we known that

$\nu\in
co(\lambda)\quad\Leftrightarrow\quad\lambda^{+}-\nu^{+}\in\bigoplus_{i=1}^{N}\mathbb{R}_{+}\alpha_{i}$

and here

$\lambda^{+}-\nu^{+}=\lambda-\nu^{+}=(\lambda_{1}-\nu^{+}_{1})(e_{1}-e_{2})+(\nu^{+}_{3}-\lambda_{3})(e_{2}-e_{3})$

Then

$\nu\in
co(\lambda)\quad\Leftrightarrow\quad\nu^{+}_{1}\leq\lambda_{1}\quad\text{and}\quad\nu^{+}_{3}\geq\lambda_{3},$

which proves the proposition, since
$\nu^{+}_{1}=\max(\nu_{1},\nu_{2},\nu_{3})$ and
$\nu^{+}_{3}=\min(\nu_{1},\nu_{2},\nu_{3})$. ∎

###  2.c Bessel function of type $A_{3}$

Let $\mu,\lambda\in C$, the Weyl chamber. We have

$\displaystyle|\overline{\mu}|$ $\displaystyle=$
$\displaystyle\mu_{1}+\mu_{2}+\mu_{3}-3\mu_{4},$
$\displaystyle\pi_{4}(\overline{\mu})$ $\displaystyle=$
$\displaystyle(\mu_{1}+\frac{\mu_{4}}{3},\mu_{2}+\frac{\mu_{4}}{3},\mu_{3}+\frac{\mu_{4}}{3}),$
$\displaystyle\pi_{4}(\nu)$ $\displaystyle=$
$\displaystyle(\frac{2\nu_{1}-\nu_{2}-\nu_{3}}{3},\frac{2\nu_{2}-\nu_{1}-\nu_{3}}{3},\frac{2\nu_{3}-\nu_{1}-\nu_{2}}{3}).$

Taking (2.3) with the change of variables

$\displaystyle z_{1}$ $\displaystyle=$
$\displaystyle\frac{\nu_{1}+\nu_{2}+\nu_{3}}{3},$ $\displaystyle x_{1}$
$\displaystyle=$ $\displaystyle\frac{2\nu_{1}-\nu_{2}-\nu_{3}}{3},$
$\displaystyle x_{2}$ $\displaystyle=$
$\displaystyle\frac{2\nu_{2}-\nu_{1}-\nu_{3}}{3}$

and put $x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$, $x_{1}+x_{2}+x_{3}=0$, we
have

$\displaystyle
J_{k,4}(\mu,\lambda)=\int_{\mathbb{R}^{3}}e^{(\mu_{1}+\mu_{2}+\mu_{3}-3\mu_{4})z_{1}}J_{k,3}(\pi_{3}(\overline{\mu}),x)V_{3}(x)$
$\displaystyle\Pi_{4}(x_{1}+z_{1},x_{2}+z_{1},x_{3}+z_{1},\lambda)\chi_{[\lambda_{2},\lambda_{1}]}(x_{1}+z_{1})\chi_{[\lambda_{3},\lambda_{2}]}(x_{2}+z_{1})\chi_{[\lambda_{4},\lambda_{3}]}(x_{3}+z_{1})dz_{1}dx_{1}dx_{2}.$

By inserting (2.6)

$\displaystyle J_{k,4}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{5}}e^{\mu_{1}+\mu_{2}+\mu_{3}-3\mu_{4})z_{1}+(\mu_{1}-\mu_{3})z_{2}+(\mu_{2}-\mu_{3})z_{3}}\delta_{k,2}((z_{2},z_{3},-(z_{2}+z_{3})),x)V_{3}(x)$
$\displaystyle\Pi_{4}(x_{1}+z_{1},x_{2}+z_{1},x_{3}+z_{1},\lambda)\chi_{[\lambda_{2},\lambda_{2}]}(x_{1}+z_{1})\chi_{[\lambda_{3},\lambda_{2}]}(x_{2}+z_{1})\chi_{[\lambda_{4},\lambda_{3}]}(x_{3}+z_{1})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
dz_{1}dz_{2}dz_{3}dx_{1}dx_{2}.$

Now with the change of variables

$\displaystyle Z_{1}$ $\displaystyle=$ $\displaystyle z_{1}+z_{2},$
$\displaystyle Z_{2}$ $\displaystyle=$ $\displaystyle z_{1}+z_{3},$
$\displaystyle Z_{3}$ $\displaystyle=$ $\displaystyle z_{1}-(z_{2}+z_{3})$

and with $Z=(Z_{1},Z_{2},Z_{3},Z_{4})\in\mathbb{R}^{4}$, such that
$Z_{1}+Z_{2}+Z_{3}+Z_{4}=0$ we have that

$(\mu_{1}+\mu_{2}+\mu_{3}-3\mu_{4})z_{1}+(\mu_{1}-\mu_{3})z_{2}+(\mu_{2}-\mu_{3})z_{3}=\mu_{1}Z_{1}+\mu_{2}Z_{2}+\mu_{3}Z_{3}+\mu_{4}Z_{4}=\langle\mu,Z\rangle.$

Therefore we can write

$\displaystyle
J_{k,3}(\mu,\lambda)=\int_{\mathbb{R}^{3}}e^{\langle\mu,Z\rangle}\delta_{k,3}(Z,\lambda)dZ_{1}dZ_{2}dZ_{3},$

where

$\displaystyle\delta_{k,3}(Z,\lambda)=$
$\displaystyle\int_{\mathbb{R}^{2}}\Pi_{4}(x_{1}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}),x_{2}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}),x_{3}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}),\lambda)$
$\displaystyle\delta_{k,2}(\frac{1}{3}(2Z_{1}-Z_{2}-Z_{3}),\frac{1}{3}(2Z_{2}-Z_{1}-Z_{3}),\frac{1}{3}(2Z_{3}-Z_{1}-Z_{2}),x)\chi_{[\lambda_{2},\lambda_{1}]}(x_{1}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}))$
$\displaystyle\chi_{[\lambda_{3},\lambda_{2}]}(x_{2}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}))\chi_{[\lambda_{4},\lambda_{3}]}(x_{3}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3}))dx_{1}dx_{2}.$
(2.7)

Let us now describe the support of $\delta_{k,3}$. In fact,
$\delta_{k,3}(Z,\lambda)\neq 0$ if the variables $x$ and $Z$ of the integrant
(2.c) satisfy:

$\displaystyle(1)$ $\displaystyle\quad\lambda_{2}\leq
x_{1}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3})\leq\lambda_{1},$ $\displaystyle(2)$
$\displaystyle\quad\lambda_{3}\leq
x_{2}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3})\leq\lambda_{2},$ $\displaystyle(3)$
$\displaystyle\quad\lambda_{4}\leq
x_{3}+\frac{1}{3}(Z_{1}+Z_{2}+Z_{3})\leq\lambda_{3},$ $\displaystyle(4)$
$\displaystyle\quad x_{3}\leq\frac{1}{3}(2Z_{1}-Z_{2}-Z_{3})\leq x_{1},$
$\displaystyle(5)$ $\displaystyle\quad
x_{3}\leq\frac{1}{3}(2Z_{2}-Z_{1}-Z_{3})\leq x_{1},$ $\displaystyle(6)$
$\displaystyle\quad x_{3}\leq\frac{1}{3}(2Z_{3}-Z_{1}-Z_{2})\leq x_{1}.$

It Follows that

$\displaystyle(1)+(4)\qquad$ $\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}\leq\lambda_{1},$ $\displaystyle(1)+(5)\qquad$ $\displaystyle\Rightarrow$
$\displaystyle\qquad Z_{2}\leq\lambda_{1},$ $\displaystyle(1)+(6)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad Z_{3}\leq\lambda_{1},$
$\displaystyle(1)+(2)+(3)\qquad$ $\displaystyle\Rightarrow$
$\displaystyle\qquad Z_{4}\leq\lambda_{1},$ $\displaystyle(1)+(2)-(6)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{2}\leq\lambda_{1}+\lambda_{2},$ $\displaystyle(1)+(2)-(5)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{3}\leq\lambda_{1}+\lambda_{2},$ $\displaystyle(1)+(2)-(4)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{2}+Z_{3}\leq\lambda_{1}+\lambda_{2},$ $\displaystyle(2)+(3)-(4)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{4}\leq\lambda_{1}+\lambda_{2},$ $\displaystyle(2)+(3)-(5)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{2}+Z_{4}\leq\lambda_{1}+\lambda_{2}$ $\displaystyle(2)+(3)-(6)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{3}+Z_{4}\leq\lambda_{1}+\lambda_{2},$ $\displaystyle(1)+(2)+(3)\qquad$
$\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{2}+Z_{3}\leq\lambda_{1}+\lambda_{2}+\lambda_{3},$
$\displaystyle(3)+(4)\qquad$ $\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{2}+Z_{3}+Z_{4}\leq\lambda_{1}+\lambda_{2}+\lambda_{3},$
$\displaystyle(3)+(5)\qquad$ $\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{3}+Z_{4}\leq\lambda_{1}+\lambda_{2}+\lambda_{3},$
$\displaystyle(3)+(6)\qquad$ $\displaystyle\Rightarrow$ $\displaystyle\qquad
Z_{1}+Z_{2}+Z_{4}\leq\lambda_{1}+\lambda_{2}+\lambda_{3}.$

These inequalities can be expressed in terms of
$Z^{+}=(Z^{+}_{1},Z^{+}_{2},Z^{+}_{3},Z^{+}_{4})$ as

$\displaystyle Z^{+}_{1}=\max(Z_{1},Z_{2},Z_{3},Z_{4})\leq\lambda_{1}$
$\displaystyle
Z^{+}_{1}+Z_{2}^{+}=\max(Z_{1}+Z_{2},Z_{1}+Z_{3},Z_{1}+Z_{4},Z_{2}+Z_{3},Z_{2}+Z_{4},Z_{3}+Z_{4})\leq\lambda_{1}+\lambda_{2}$
$\displaystyle
Z^{+}_{1}+Z_{2}^{+}+Z^{+}_{3}=\max(Z_{1}+Z_{2}+Z_{3},Z_{2}+Z_{3}+Z_{4},Z_{1}+Z_{2}+Z_{4},Z_{1}+Z_{3}+Z_{4})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq\lambda_{1}+\lambda_{2}+\lambda_{3}$

which imply that $Z^{+}\preceq\lambda$ and therefore $Z\in co(\lambda)$.

###  2.d Case for arbitrary $N$

After having idea about the case $N=2,3$ it is not hard to see that the
formula (1.1) can be found using recurrence. In fact, let $\mu,\lambda\in C$
the Weyl chamber. Put for $\nu\in\mathbb{R}^{N-1}$

$\Omega(\lambda,\nu)=\prod_{i=1}^{N-1}\chi_{[\lambda_{i+1},\lambda_{i}]}(\nu).$

With the change of variables

$\displaystyle z_{1}=\frac{|\nu|}{N-1}=\frac{\nu_{1}+...+\nu_{N-1}}{N-1}$
$\displaystyle x_{i}=\nu_{i}-\frac{|\nu|}{N-1};\qquad 1\leq i\leq N-2$

the formula (2.3) becomes,

$\displaystyle J_{k,N}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\mathbb{R}^{N-1}}e^{|\overline{\mu}|z_{1}}J_{k,N-1}(\pi_{N-1}(\overline{\mu}),x)V_{N-1}(x)$
$\displaystyle\Pi_{N}(\lambda,(x_{1}+z_{1},...,x_{N-1}+z_{1}))\Omega(\lambda,(x_{1}+z_{1},...,x_{N-1}+z_{1}))dz_{1}dx_{1}...dx_{N-2}$

where we put $x=(x_{1},...,x_{N-1})$ with $x_{N-1}=-(x_{1}+...+x_{N-2})$. The
recurrence hypothesis says that

$\displaystyle J_{k,N-1}(\pi_{N-1}(\overline{\mu}),x)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{V}_{N-1}}e^{\langle\pi_{N-1}(\overline{\mu}),z\rangle}\delta_{k,N-1}(x,z)dz.$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N-2}}e^{\sum_{i=1}^{N-1}\left(\overline{\mu}_{i}-\frac{|\overline{\mu}|}{N-1}\right)z_{i+1}}\delta_{k,N-1}(x,z)dz_{2}...dz_{N-2}.$

where $z=(z_{2},...,z_{N})$ with $z_{N}=-(z_{2}+...+z_{N-1})$ and
$\delta_{k,N-1}(.,x)$ is supported in the convex hull of $S_{N-1}.x$ in
$\mathbb{R}^{N-1}$. Hence we get

$\displaystyle J_{k,N}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\mathbb{R}^{N-1}}\int_{\mathbb{R}^{N-2}}e^{|\overline{\mu}|z_{1}+\sum_{i=1}^{N-1}\left(\overline{\mu}_{i}-\frac{|\overline{\mu}|}{N-1}\right)z_{i+1}}\delta_{k,N-1}(z,x)$
$\displaystyle
V_{N-1}(x)\Pi_{N}(\lambda,(x_{1}+z_{1},...,x_{N-1}+z_{1}))\Omega(\lambda,(x_{1}+z_{1},...,x_{N-1}+z_{1}))$
$\displaystyle\qquad\qquad\qquad\qquad
dz_{1}dz_{2}...dz_{N-1}dx_{1}...dx_{N-2}.$

Now observing that

$\displaystyle|\overline{\mu}|z_{1}+\sum_{i=1}^{{}^{N-1}}\left(\overline{\mu}_{i}-\frac{|\overline{\mu}|}{N-1}\right)z_{i+1}$
$\displaystyle=$
$\displaystyle\left(\sum_{i=1}^{N-1}\mu_{i}-(N-1)\mu_{N}\right)z_{1}+\sum_{i=1}^{N-1}\mu_{i}z_{i+1}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N-1}\mu_{i}(z_{1}+z_{i+1})-(N-1)\mu_{N-1}z_{1}.$

Then making the change of variables

$\displaystyle Z_{i}=z_{1}+z_{i+1},\qquad 1\leq i\leq N-1$

and put $Z=(Z_{1},...,Z_{N})$ with $Z_{N}=-(Z_{1}+...+Z_{N-1}$, we have

$\displaystyle J_{k,N}(\mu,\lambda)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\mathbb{R}^{N-1}}\int_{\mathbb{R}^{N-2}}e^{\sum_{i=1}^{N}\mu_{i}Z_{i}}\delta_{k,N-1}(\phi(Z),x)V_{N-1}(x)\Pi_{N}(\lambda,\theta(Z,x))$
$\displaystyle\qquad\qquad\qquad\Omega_{N}(\lambda,\theta(Z,x))dZ_{1}dZ_{2}...dZ_{N-1}dx_{1}...dx_{N-2}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N-1}}e^{\sum_{i=1}^{N}\mu_{i}Z_{i}}\delta_{k,N}(\lambda,Z)dZ_{1}...dZ_{N-1}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{V}_{N}}e^{\langle\mu,Z\rangle}\delta_{k,N}(\lambda,Z)dZ,$

with

$\displaystyle\phi(Z)$ $\displaystyle=$
$\displaystyle\left(Z_{1}-\frac{\sum_{i=1}^{N-1}Z_{i}}{N-1},...,Z_{N_{-1}}-\frac{\sum_{i=1}^{N-1}Z_{i}}{N-1}\right)$
$\displaystyle\theta(Z,x)$ $\displaystyle=$
$\displaystyle\left(x_{1}+\frac{\sum_{i=1}^{N-1}Z_{i}}{N-1},...,x_{N-1}+\frac{\sum_{i=1}^{N-1}Z_{i}}{N-1}\right)$
$\displaystyle\delta_{k,N}(\lambda,Z)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Nk)}{V_{N}(\lambda)^{2k-1}\Gamma(k)^{N}}\int_{\mathbb{R}^{N-2}}\delta_{k,N-1}(\phi(Z),x)$
(2.8) $\displaystyle\qquad\qquad\qquad
V_{N-1}(x)\Pi_{N}(\lambda,\theta(Z,x))\Omega_{N}(\lambda,\theta(Z,x))dx_{1}...dx_{N-2}.$

Now we write sufficient conditions for which the integrant (2.8) does not
vanish

$\displaystyle(\Lambda_{i})$ $\displaystyle\qquad\lambda_{i+1}\leq
x_{i}+\frac{\sum_{i=1}^{N-1}Z_{i}}{N-1}\leq\lambda_{i}$
$\displaystyle(\Lambda_{I})$ $\displaystyle\qquad\sum_{i\in
I}Z_{i}-\frac{|I|}{N-1}\sum_{i=1}^{N-1}Z_{i}\leq\sum_{i=1}^{|I|}x_{i}$

for all $I\subset\\{1,2,...,N-1\\}$ of cardinally $|I|$. It follows that

$\sum_{i=1}^{|I|}\Lambda_{i}+\Lambda_{I}\quad\Rightarrow\quad\sum_{i\in
I}Z_{i}\leq\sum_{i=1}^{|I|}\lambda_{i}$

which proves that $Z^{+}\leq\lambda$ and then $Z\in co(\lambda)$.

##  3 Partially product formula for $J_{k}$

We will first establish a product formula for $J_{k}$ provided that a
conjecture of Stanley on the multiplication of Jack polynomials is true. The
conjecture says that for all partitions $\mu$ and $\lambda$

$\displaystyle
j_{\mu}j_{\lambda}=\sum_{\nu\leq\mu+\lambda}g_{\mu,\lambda}^{\nu}j_{\nu}$

where $g_{\mu,\lambda}^{\nu}$ ( the Littlewood-Richardson coefficients ) is a
polynomial in $k$ with nonnegative integer coefficients. In particular,
$g_{\mu,\lambda}^{\nu}\geq 0$, what is the interesting facts in our setting.
Hence we have for all $\mu,\lambda$ partitions,

$\displaystyle
F(\pi(\mu)+\rho_{k},.)F(\pi(\lambda)+\rho_{k},.)=\sum_{\nu\leq\mu+\lambda}f_{\mu,\lambda}^{\nu}F(\pi(\nu)+\rho_{k},.)$

with $f_{\mu,\lambda}^{\nu}\geq 0$ and

$\sum_{\nu}f_{\mu,\lambda}^{\nu}=1$

But if $\nu\leq\mu+\lambda$ as partitions then we also have
$\pi(\nu)\preceq\pi(\mu)+\pi(\lambda)$ in the dominance ordering ([2], Lemma
3.1 ). This allows us to write for all $\mu,\lambda\in P^{+}$

$\displaystyle F(\mu+\rho_{k},.)F(\lambda+\rho_{k},.)=\sum_{\nu\in
P^{+};\;\nu\preceq\mu+\lambda}f_{\mu,\lambda}^{\nu}F(\nu+\rho_{k},.).$

To arrive at product formula for $J_{k}$ we follow the technic used by M.
Rösler in [14]. We first write

$F(n\mu+\rho_{k},\frac{z}{n})F(n\lambda+\rho_{k},\frac{z}{n})=\int_{\mathbb{R}^{N}}F(nx+\rho_{k},\frac{z}{n})d\gamma_{\mu,\lambda}^{n}(x),\qquad
z\in\mathbb{V}.$

where

$d\gamma_{\mu,\lambda}^{n}=\sum_{\nu\in
P^{+};\;\nu\preceq\mu+\lambda}f_{\mu,\lambda}^{\nu}\;\delta_{\frac{\nu}{n}}.$

According to ([14], Lemma 3.2) the probability measure
$\gamma_{\mu,\lambda}^{n}$ is supported in the convex hull $co(\mu+\lambda)$.
So, from Prohorov’s theorem (see [3] ) there exists a probability measure
$\gamma_{\mu,\lambda}$ supported in $co(\mu+\lambda)$ and a subsequence
$(\gamma_{\mu,\lambda}^{n_{j}})_{j}$ which converges weakly to
$\gamma_{\mu,\lambda}$. Then by using (1.6) it follows that

$J_{k}(\mu,z)J_{k}(\lambda,z)=\int_{\mathbb{V}}J_{k}(\xi,z)d\gamma_{\mu,\lambda}(\xi)$

for all $z\in\mathbb{V}$ and $\mu,\lambda\in P^{+}$.
Now let $r,s\in\mathbb{Q}^{+}$ with $r=\frac{a}{b}$ and $s=\frac{c}{b}$,
$a,b,c\in\mathbb{N}$, $b\neq 0$. We write

$\displaystyle J_{k}(r\mu,z)J_{k}(s\lambda,z)$ $\displaystyle=$ $\displaystyle
J_{k}(a\mu,\frac{z}{b})J_{k}(c\lambda,\frac{z}{b})$ $\displaystyle=$
$\displaystyle\int_{\mathbb{V}}J_{k}(\xi,\frac{z}{b})d\gamma_{a\mu,c\lambda}(\xi);\quad
z\in\mathbb{R}^{d}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{V}}J_{k}(\frac{\xi}{b},z)d\gamma_{a\mu,c\lambda}(\xi);\quad
z\in\mathbb{R}^{d}$

Defining $\gamma_{r\mu,s\lambda}$ as the image measure of
$\gamma_{a\mu,c\lambda}$ under the dilation $\xi\rightarrow\frac{\xi}{b}$. We
get

$J_{k}(r\mu,z)J_{k}(s\lambda,z)=\int_{\mathbb{V}}J_{k}(\xi,z)d\gamma_{r\mu,s\lambda}(\xi).$

Now we apply the density argument, since
$\mathbb{Q}^{+}.P^{+}\times\mathbb{Q}^{+}.P^{+}$ is dense in $C\times C$,
where $C$ is the Weyl chamber. Then Prohorov’s theorem yields

$\displaystyle
J_{k}(\mu,z)J_{k}(\lambda,z)=\int_{\mathbb{V}}J_{k}(\xi,z)d\gamma_{\mu,\lambda}(\xi);\quad
z\in\mathbb{R}^{d}$

for all $\mu,\lambda\in C$ with $supp(\gamma_{\mu,\lambda})\subset
co(\mu+\lambda)$. This finish our approach for the product formula.

An important special case of the Stanley conjecture called Peiri formula is
where the partition $\lambda=(n)$, $n\in\mathbb{N}$. Since this formula has
already been proved (see [16]) then we can state the following partial result

######  Theorem 2.

For all $\mu\in C$ and all $t\geq 0$ there exists a probability measure
$\gamma_{\mu,t}$ such that

$\displaystyle
J_{k}(\mu,z)J_{k}(t\beta_{1},z)=\int_{\mathbb{V}}J_{k}(\xi,z)d\gamma_{\mu,t}(\xi);\quad
z\in\mathbb{R}^{d}$

where $\beta_{1}=\pi(e_{1})$. The measure $\gamma_{\mu,t}$ is supported in
$co(\mu+t\beta_{1})$.

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