# The Dunkl-Laplace transform and Macdonald’s hypergeometric series

Dominik Brennecken and Margit Rösler  Institut für Mathematik, Universität
Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We continue a program started in [Rös20], generalizing classical results from
the analysis on symmetric cones to the Dunkl setting for root systems of type
$A$. In particular, we prove a Dunkl-Laplace transform identity for Heckman-
Opdam hypergeometric functions of type $A$ and more generally, for the
associated Cherednik kernel. This is achieved by analytic continuation from a
Laplace transform identity for non-symmetric Jack polynomials which was
stated, for the symmetric case, as a key conjecture by Macdonald in [Mac13].
Our proof for the Jack polynomials is based on Dunkl operator techniques and
the raising operator of Knop and Sahi. Moreover, we use these results to
establish Laplace transform identities between hypergeometric series in terms
of Jack polynomials. Finally, we conclude with a Post-Widder inversion formula
for the Dunkl-Laplace transform.

###### Key words and phrases:

Dunkl theory, Laplace transform, Jack polynomials, Heckman-Opdam theory,
hypergeometric functions

###### 2020 Mathematics Subject Classification:

Primary 33C67; Secondary 05E05, 33C52, 33C80, 44A10

##  1\. Introduction

The Laplace transform is an important tool in various areas of harmonic
analysis and forms a cornerstone in the analysis on symmetric cones, see
[FK94]. In particular, there are important Laplace transform identities
between $\,{}_{p}F_{q}$-hypergeometric functions on a symmetric cone, which
are given as expansions with respect to the associated spherical polynomials,
c.f. [FK94, Chap.XV]. For cones of positive definite matrices, such
hypergeometric series trace back to ideas of Bochner and were studied in
detail by Herz [Her55], where they were actually defined recursively by means
of the Laplace transform. For important further developments see for instance
[Con63, GR89, Kan93]. Multivariable hypergeometric series have found many
applications in multivariate statistics [Mui82], but also in number theory and
mathematical physics.

In his unpublished manuscript [Mac13] from the 1980ies, Macdonald introduced
hypergeometric series in terms of Jack polynomials, which include the
hypergeometric functions on symmetric cones as special cases. In this context,
he also introduced a generalization of the Laplace transform for radial
functions on symmetric cones, but many statements in [Mac13] remained at a
formal and conjectural level.

Radial analysis on symmetric cones is closely related to Dunkl theory for root
systems of type $A$, and also Macdonald’s concepts have a natural
interpretation within Dunkl theory, because the ${}_{0}F_{0}$-hypergeometric
function, which replaces the exponential kernel in the Macdonald’s Laplace
transform, is just a Dunkl-Bessel function of type $A$. The connection of the
concepts in [Mac13] to Dunkl theory was already observed by Baker and
Forrester in their seminal papers [BF97, BF98] related to the study of
Calogero-Moser-Sutherland models.

To become more precise on these connections, consider a symmetric cone
$\Omega=G/K$ inside a simple Euclidean Jordan algebra $V$ of rank $n$ and with
Peirce constant $d.$ Let $F\in L^{1}_{loc}(\Omega)$ be $K$-invariant, that is
of the form $F(x)=f(\text{spec}(x)),$ where
$\text{spec}(x)\in\mathbb{R}_{+}^{n}$ with $\mathbb{R}_{+}=]0,\infty[$ denotes
the set of eigenvalues of $x$ ordered by size. Then for $y\in\Omega$, the
Laplace transform

$LF(y)=\int_{\Omega}e^{-(x|y)}F(x)dx$

depends only on $\eta=\text{spec}(y)\in\mathbb{R}_{+}^{n}$ and can be written
as

$LF(y)=\text{const}\cdot\int_{\mathbb{R}_{+}^{n}}J(-\xi,\eta)f(\xi)\omega(\xi)d\xi,$
(1.1)

with

$J(-\xi,\eta)=\int_{K}e^{-(x|ky)}dk,\quad\xi=\text{spec}(x),\eta=\text{spec}(y),$

and

$\omega(\xi)=\prod_{i<j}|\xi_{i}-\xi_{j}|^{d}.$

It is well-known (c.f. [Rös20] for details) that

$J(\xi,\eta)=\sum_{\lambda\in\Lambda_{n}^{+}}\frac{1}{|\lambda|!}\frac{C_{\lambda}(\xi;\tfrac{2}{d})C_{\lambda}(\eta;\tfrac{2}{d})}{C_{\lambda}(\underline{1};\tfrac{2}{d})},$
(1.2)

where $\Lambda_{n}^{+}$ denotes the set of partitions with at most $n$ parts
and the $C_{\lambda}(\,.\,;\alpha)$ are the (symmetric) Jack polynomials of
index $\alpha\geq 0$ in $C$-normalization as in Lemma 6.1. On the other hand,
the kernel $J$ coincides with a Dunkl-Bessel function of type $A_{n-1}$ with
multiplicity $k=d/2.$ Indeed, the Dunkl-Bessel function $J_{k}$ associated
with root system $A_{n-1}$ and arbitrary multiplicity $k\geq 0$ has an
expansion as a ${}_{0}F_{0}$-hypergeometric function in the sense of [Mac13,
Kan93]:

$J_{k}(z,w)=\sum_{\lambda\in\Lambda_{n}^{+}}\frac{1}{|\lambda|!}\frac{C_{\lambda}(z;\alpha)C_{\lambda}(w;\alpha)}{C_{\lambda}(\underline{1};\alpha)}=\,_{0}F_{0}^{\alpha}(z,w),\quad\alpha=\frac{1}{k}\in\,]0,\infty],$

see Remark 6.6. Macdonald [Mac13] considered the Laplace transform (1.1) with
the Bessel function $e(z,w)=\,_{0}F_{0}^{\alpha}(z,w)$ for arbitrary indices
$\alpha>0$ instead of the specific values $\alpha=2/d$ only. Many of his
calculations were of a formal nature and rested on a conjecture about the
Laplace transform of Jack polynomials, named Conjecture (C), which can be
formulated as follows: Let

$\mu_{0}:=k(n-1)\,\text{ and }\,\Delta(z):=\prod_{j=1}^{n}z_{j}\text{ for
}\,z\in\mathbb{C}^{n}$

and write $C_{\lambda}(z):=C_{\lambda}(z;\tfrac{1}{k})$ for abbreviation. Then
for all $\lambda\in\Lambda_{n}^{+},\,y\in\mathbb{R}_{+}^{n}$ and all
$\mu\in\mathbb{C}$ with $\text{Re}\,\mu>\mu_{0}\,,$

$\int_{\mathbb{R}_{+}^{n}}J_{k}(-y,x)\,C_{\lambda}(x)\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx\,=\,\Gamma_{n}(\lambda+\underline{\mu})\,C_{\lambda}\bigl{(}\tfrac{1}{y}\bigr{)}\,\Delta(y)^{-\mu}.$
(1.3)

Here $\Gamma_{n}(\lambda)=\Gamma_{n}(\lambda;k)$ is a multivariate gamma
function defined in formula (3.7) below. In [Rös20], we gave a rigorous
treatment of the Dunkl-Laplace transform

$\mathcal{L}f(z)=\int_{\mathbb{R}_{+}^{n}}E_{k}(-z,x)f(x)\omega(x)dx$

where compared to Macdonald’s version, the Bessel function is replaced by the
Dunkl kernel $E_{k}$ of type $A_{n-1}$ and multiplicity $k.$ This transform
was already considered by Baker and Forrester [BF98] and later used in [SZ07],
but just as for Macdonald’s transform, convergence issues had remained open
for a long time.

Formula (1.3) generalizes a Laplace transform identity for spherical
polynomials on a symmetric cone, which is in turn a consequence of the
following important Laplace transform identity for the generalized power
functions $\Delta_{s}$, $s=(s_{1},\ldots,s_{n})\in\mathbb{C}^{n}$ with
$\text{Re}\,s_{j}>\frac{d}{2}(j-1)$ (see [FK94, Chapter VII]):

$\int_{\Omega}e^{-(y|x)}\Delta_{s}(x)\Delta(x)^{-m/n}dx\,=\,\Gamma_{\Omega}(s)\Delta_{s}(y^{-1})\quad(y\in\Omega),$
(1.4)

where $m$ is the dimension of the Jordan algebra, $\Delta$ denotes the Jordan
determinant and $\Gamma_{\Omega}$ is the Gindikin gamma function associated
with $\Omega$. Taking $K$-means in (1.4), one gets a Laplace transform
identity for the spherical functions of $\Omega$, which may be parametrized as

$\varphi_{\lambda}(x)=\int_{K}\Delta_{\lambda}(kx)dk,\quad\lambda\in\mathbb{C}^{n}$

and depend only on the eigenvalues of their argument. More precisely, for
$y\in\Omega$ and $s\in\mathbb{C}^{n}$ as above,

$\int_{\Omega}e^{-(y|x)}\varphi_{s}(x)\Delta(x)^{-m/n}dx\,=\,\Gamma_{\Omega}(s)\varphi_{s}(y^{-1}).$
(1.5)

It is well-known that the spherical functions of $\Omega$ can be expressed in
terms of Heckman-Opdam hypergeometric functions of type $A_{n-1}$ and with
multiplicity $k=d/2,$ see e.g. [RKV13] for cones of positive definite matrices
over $\mathbb{R},\mathbb{C},\mathbb{H}.$ For parameters
$\lambda\in\Lambda_{n}^{+}$, the $\varphi_{\lambda}$ are just the spherical
polynomials of $\Omega$ given by

$\varphi_{\lambda}(x)=\frac{C_{\lambda}(\text{spec}(x),\tfrac{2}{d})}{C_{\lambda}(\underline{1},\tfrac{2}{d})}.$

Rewriting (1.5) by means of identity (1.1), one gets formula (1.3) for the
particular multiplicities $k=d/2.$

In the present paper, we shall establish a generalization of formula (1.5) to
the Dunkl setting of type $A_{n-1}$ with arbitrary nonnegative multiplicity.
Namely, we obtain in Theorem 5.3 a Dunkl-Laplace transform identity for
Heckman-Opdam hypergeometric functions of type $A_{n-1}$ and, more generally,
for the associated Opdam-Cherednik kernel. The first step will be a rigorous
proof of Macdonald’s Conjecture (C). More generally, we shall prove Dunkl-
Laplace transform identities for the non-symmetric Jack polynomials in the
sense of [Opd95, KS97], from which (1.3) then follows by symmetrization. These
non-symmetric identities were already stated in [BF98], but the proof given
there in terms of Laguerre expansions is involved and not fully carried out.
The proof we are presenting here is completely different and very natural; it
is based on a reformulation via Dunkl operators and is carried out by
induction, using the raising operator of Knop and Sahi [KS97] for the non-
symmetric Jack polynomials. The statement for the Cherednik kernel is then
obtained via analytic continuation with respect to the spectral variable, and
for the hypergeometric function it follows by symmetrization.

Based on the Laplace transform identities for Jack polynomials, we then study
hypergeometric series (in two variables) in terms of Jack polynomials and
establish Laplace transform identities between them. This generalizes known
results on symmetric cones and settles several conjectural Laplace transform
formulas in [Mac13]. As a further application, we finally prove a Post-Widder
inversion theorem for the Dunkl-Laplace transform, which complements a result
by Faraut and Gindikin in [FG90] for the Laplace transform on symmetric cones.

The organization of this paper is as follows: Section 2 provides the necessary
background on the type $A$ Dunkl setting, both in the rational and
trigonometric case. In Section 3, we collect results on the symmetric and non-
symmetric Jack polynomials which will be relevant in the sequel, and we prove
the Dunkl-Laplace transform identities for Jack polynomials. Section 4
contains a digression on some useful properties of the Cherednik kernel for
arbitrary root systems. These will be employed (for type $A_{n-1}$) in Section
5, where the Laplace transform identities for Jack polynomials are extended to
the Cherednik kernel and to the hypergeometric function. Section 6 is devoted
to the study of Jack-hypergeometric series, and Section 7 contains the Post-
Widder inversion formula in the Dunkl setting.

To avoid notational overload, we shall always suppress in our notations the
dependence on the fixed multiplicity parameter $k\geq 0$ on the root system
$A_{n-1}$.

##  2\. The type A Dunkl setting

For a general background, the reader is referred to [Dun89, dJ93, Rös03, DX14,
Opd95, Rös20, HO21].

We consider the root system $A_{n-1}=\\{\pm(e_{i}-e_{j}):1\leq i<j\leq n\\}$
in the Euclidean space $\mathbb{R}^{n}$ with inner product $\langle
x,y\rangle=\sum_{i=1}^{n}x_{i}y_{i}$ and norm $|x|=\sqrt{\langle x,x\rangle}$,
where the $e_{i}$ denote the standard basis vectors. The inner product
$\langle\cdot,\cdot\rangle$ is extended to $\mathbb{C}^{n}$ in a bilinear way.
The reflection group generated by $A_{n-1}$ is the symmetric group
$\mathcal{S}_{n}$ on $n$ elements. It acts on functions
$f:\mathbb{R}^{n}\to\mathbb{C}$ by $\sigma f:=f(\sigma^{-1}\cdot).$ The
(rational) Dunkl operators associated with $A_{n-1}$ and a multiplicity
parameter $k\in\mathbb{C}$ are given by

$T_{\xi}=T_{\xi}(k)=\partial_{\xi}+\frac{k}{2}\cdot\\!\sum_{\alpha\in
A_{n-1}}\langle\alpha,\xi\rangle\frac{1-s_{\alpha}}{\langle\alpha,x\rangle}\quad(\xi\in\mathbb{R}^{n})$

where $s_{\alpha}$ denotes the orthogonal reflection in the root $\alpha$. For
fixed $k$, the operators $T_{\xi}(k),\,\xi\in\mathbb{R}^{n}$ commute. Let
$\mathcal{P}:=\mathbb{C}[\mathbb{R}^{n}]$ denote the space of polynomial
functions on $\mathbb{R}^{n}$. The assignment
$\,\langle\cdot,\xi\rangle\mapsto T_{\xi}$ has a unique extension to a unital
morphism of algebras $\phi:\mathcal{P}\to\text{End}(\mathcal{P})$, and we
shall write $\phi(p)=:p(T).$

The Dunkl operators satisfy a product rule: if $f,g\in C^{1}(\mathbb{R}^{n})$
and one of them is symmetric, then $T_{\xi}(fg)=(T_{\xi}f)g+fT_{\xi}g.$

In this paper, we shall always assume that $k\geq 0.$ Let
$E(\lambda,\cdot):=E_{k}(\lambda,\cdot):\mathbb{C}^{n}\to\mathbb{C}$,
$\lambda\in\mathbb{C}^{n}$ be the associated Dunkl kernel of type $A_{n-1},$
i.e. the unique holomorphic solution of the system

$\begin{cases}T_{\xi}E(\lambda,\cdot)=\braket{\lambda,\xi}E(\lambda,\cdot),\quad\text{
for all }\xi\in\mathbb{R}^{n}\\\ \;\,\;E(\lambda,0)=1\end{cases}.$

The associated Bessel function is defined by

$J(\lambda,z):=J_{k}(\lambda,z):=\frac{1}{n!}\sum_{\sigma\in\mathcal{S}_{n}}E(\lambda,\sigma
z).$

The Dunkl kernel is positive on $\mathbb{R}^{n}\times\mathbb{R}^{n}$ and for
$\lambda,z\in\mathbb{C}^{n},s\in\mathbb{C},\sigma\in\mathcal{S}_{n}$ one has

$E(\lambda,z)=E(z,\lambda),\quad E(s\lambda,z)=E(\lambda,sz),\quad
E(\sigma\lambda,\sigma z)=E(\lambda,z).$

Moreover, it satisfies

$E(\lambda,z+\underline{s})=e^{\langle\lambda,\underline{s}\rangle}\cdot
E(\lambda,z)$

for $s\in\mathbb{C},$ where

$\underline{s}:=s\cdot(1,\ldots,1)\in\mathbb{C}^{n}.$

Let

$\omega(x)=\prod_{1\leq i<j\leq n}|x_{i}-x_{j}|^{2k}.$

Dunkl analysis associated with the root system $A_{n-1}$ generalizes the
radial analysis on symmetric cones, which just corresponds to the multiplicity
values $k=d/2$, where $d$ is the Peirce constant of the cone. There is a well-
behaved Laplace transform of functions $f\in L_{loc}^{1}(\mathbb{R}_{+}^{n}),$
for arbitrary $k\geq 0$, which is given by

$\mathcal{L}f(z)=\int_{\mathbb{R}_{+}^{n}}f(x)E(-z,x)\omega(x)dx,$

and was first considered in [BF98]. For $x\in\mathbb{R}_{+}^{n}$ and
$z\in\mathbb{C}^{n}$ with $\,\text{Re}\,z\geq a$ for some $a\in\mathbb{R}^{n}$
(which is understood componentwise), the type $A$ Dunkl kernel satisfies the
exponential bound

$|E(-z,x)|\leq\,\exp\bigl{(}-\|x\|_{1}\cdot\min_{1\leq i\leq
n}a_{i}\bigr{)},\,$ (2.1)

see [Rös20]. Here $\left\lVert x\right\rVert_{1}=\sum_{i=1}^{n}|x_{i}|.\,$
This estimate, which seems to be exclusive in type $A$, guarantees good
convergence properties of the Laplace integral. In particular, we recall the
following

######  Lemma 2.1 ([Rös20]).

Suppose that $f:\mathbb{R}_{+}^{n}\to\mathbb{C}$ is measurable and
exponentially bounded according to $\,|f(x)|\leq Ce^{s\|x\|_{1}}$ with some
constants $C>0$ and $s\in\mathbb{R}$. Then $\mathcal{L}f(z)$ exists and is
holomorphic on $\\{z\in\mathbb{C}^{n}:\text{Re}\,z>\underline{s}\\}.$
Moreover, for each polynomial $p\in\mathcal{P},$
$\,p(-T)(\mathcal{L}f)=\mathcal{L}(fp)$ on $\\{\text{Re}\,z>\underline{s}\\}.$

Let us turn to the trigonometric setting. Here root systems are required to be
spanning, and therefore we consider $A_{n-1}$ as a subset of

$\mathbb{R}_{0}^{n}:=\\{x\in\mathbb{R}^{n}:x_{1}+\ldots+x_{n}=0\\}.$

We fix the positive subsystem $A_{n-1}^{+}=\\{e_{j}-e_{i}:i<j\\}.$ The
(trigonometric) Cherednik operators on $\mathbb{R}_{0}^{n}$ associated with
$A_{n-1}^{+}$ and a multiplicity parameter $k\geq 0$ are given by

$D_{\xi}=D_{\xi}(A_{n-1}^{+},k):=\partial_{\xi}-\braket{\rho(R_{+}),\xi}+k\sum\limits_{\alpha\in
A_{n-1}^{+}}\braket{\alpha,\xi}\frac{1-s_{\alpha}}{1-e^{-\braket{\,\cdot,\alpha}}},\quad\xi\in\mathbb{R}^{n},$

with the Weyl vector

$\rho(R_{+})=\rho(R_{+},k):=\,\frac{k}{2}\sum\limits_{\alpha\in
A_{n-1}^{+}}\\!\alpha=\,-\frac{k}{2}(n-1,n-3,\ldots,-n+1).$

The operators $D_{\xi}\,,\,\xi\in\mathbb{R}_{0}^{n}$ commute. Let
$\mathbb{C}_{0}^{n}=\mathbb{R}_{0}^{n}\oplus i\mathbb{R}_{0}^{n}.$ Due to
[Opd95], there exist an $\mathcal{S}_{n}$-invariant tubular open neighborhood
$U$ of $\mathbb{R}_{0}^{n}$ in $\mathbb{C}_{0}^{n}$ and a unique holomorphic
function $G=G_{k}$ on $\mathbb{C}_{0}^{n}\times(\mathbb{R}_{0}^{n}+iU)$ which
satisfies the joint eigenvalue problem

$\begin{cases}D_{\xi}G(\lambda,\cdot)=\braket{\lambda,\xi}G(\lambda,\cdot),\quad\text{
for all }\xi\in\mathbb{R}_{0}^{n}\\\ \;\;\;\,G(\lambda,0)=1.\end{cases}$

The function $G$ is called the (Opdam-) Cherednik kernel associated with
$A_{n-1}$ and multiplicity $k.$ By symmetrization, one obtains the associated
Heckman-Opdam hypergeometric function

$F(\lambda,z)=F_{k}(\lambda,z)=\frac{1}{n!}\sum\limits_{\sigma\in\mathcal{S}_{n}}G_{k}(\lambda,\sigma
z).$

It is actually $\mathcal{S}_{n}$-invariant in both $\lambda$ and $z$.
According to [KO08, Theorem 13.15] (see also [HO21, Cor. 8.6.2]), $F$ extends
to a holomorphic function on
$\mathbb{C}_{0}^{n}\times(\mathbb{R}_{0}^{n}+i\Omega)$ with

$\Omega=\\{x\in\mathbb{R}_{0}^{n}:|x_{i}-x_{j}|<\pi\quad\text{ for all }1\leq
i<j\leq n\\},$

and the proof of [Opd95, Theorem 3.15] shows that $G$ extends holomorphically
to the same domain. We mention that For $k=d/2$ with $d=1,2,4,$ the functions
$t\mapsto F(\lambda,2t)$ on $\mathbb{R}_{0}^{n}$ are naturally identified with
the spherical functions of $SL_{n}(\mathbb{F})/SU_{n}(\mathbb{F})$ with
$\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$; cf. [Opd95, RKV13].

In this paper, we shall work with natural extensions of $G$ and $F.$ In order
to define them, we consider the Cherednik operators $D_{\xi}$ as operators on
$\mathbb{R}^{n}$, for arbitrary $\xi\in\mathbb{R}^{n}.$

######  Lemma 2.2.

Consider the orthogonal projection
$\pi:\mathbb{R}^{n}\to\mathbb{R}_{0}^{n},\,x\mapsto x-\frac{1}{n}\langle
x,\underline{1}\rangle\cdot\underline{1}.$ Then for all
$\,\xi\in\mathbb{R}^{n}$ and $f,g\in C^{1}(\mathbb{R}^{n}),$

  1. (1)

$D_{\xi}(f\circ\pi)=D_{\pi(\xi)}f\circ\pi$.

  2. (2)

If $f$ or $g$ is $\mathcal{S}_{n}$-invariant, then
$D_{\xi}(fg)=f(D_{\xi}g)+(D_{\xi}f)g$.

###### Proof.

Part (1) is obtained by a short calculation, using that
$\partial_{\xi}(f\circ\pi)=\partial_{\pi(\xi)}f\circ\pi$ and that $\rho$ and
all roots are contained in $\mathbb{R}_{0}^{n}.$ Part (2) is straightforward.
∎

In order to extend the Cherednik kernel, put
$V:=\Omega+\mathbb{R}\underline{1}\subset\mathbb{R}^{n},$ which is
$\mathcal{S}_{n}$-invariant with $\pi(V)=\Omega.$ Keeping the notation, we
define

$G:\mathbb{C}^{n}\times(\mathbb{R}^{n}+iV)\to\mathbb{C},\,\,G(\lambda,z):=e^{\frac{1}{n}\langle
z,\underline{1}\rangle\langle\lambda,\underline{1}\rangle}\,G(\pi(\lambda),\pi(z)),$
(2.2)

where $\pi$ is canonically extended to a linear mapping
$\pi:\mathbb{C}^{n}\to\mathbb{C}_{0}^{n}.$ The extended hypergeometric
function $F$ is obtained in the same way.

######  Proposition 2.3.

The extended Cherednik kernel $G(\lambda,\cdot),\lambda\in\mathbb{C}^{n}$ is
the unique holomorphic solution on $\mathbb{R}^{n}+iV$ to the system

$\begin{cases}D_{\xi}G(\lambda,\cdot)=\langle\lambda,\xi\rangle
G(\lambda,\cdot)&\text{ for all }\,\xi\in\mathbb{R}^{n}\\\ \quad
G(\lambda,0)=1.\par\end{cases}$ (2.3)

###### Proof.

Since $z\mapsto e^{\langle
z,\underline{1}\rangle\langle\lambda,\underline{1}\rangle/n}\,$ is
$\mathcal{S}_{n}$-invariant, it is immediate from Lemma 2.2(2) that
$G(\lambda,\cdot)$ solves (2.3). Assume that $f$ is a further solution in some
neighborhood $U^{\prime}$ of $0\in\mathbb{C}^{n}.$ Then for
$\xi\in\mathbb{R}_{0}^{n}$, Lemma 2.2 (1) gives

$D_{\xi}(f\circ\pi)=D_{\xi}f\circ\pi=\braket{\lambda,\xi}(f\circ\pi)=\braket{\pi(\lambda),\xi}(f\circ\pi).$

The uniqueness of the Cherednik kernel on $\mathbb{R}_{0}^{n}$ thus implies
that $f$ coincides with $G(\pi(\lambda),\cdot)$ on $\pi(U^{\prime})$. Finally,
for $\xi=\underline{1}$ we have $D_{\underline{1}}=\partial_{\underline{1}}$,
and the eigenvalue equation
$D_{\underline{1}}f=\langle\lambda,\underline{1}\rangle f\,$ leads to

$\partial_{\underline{1}}\Bigl{(}e^{-\braket{\cdot,\underline{1}}\braket{\lambda,\underline{1}}/n}\cdot
f\Bigr{)}\equiv 0\quad\text{on }\mathbb{R}^{n}.$

Therefore $\,x\mapsto
e^{-\tfrac{\braket{x,\underline{1}}\braket{\lambda,\underline{1}}}{n}}f(x)$ is
in each point constant in direction $\underline{1}$, i.e.

$e^{-\tfrac{\braket{x,\underline{1}}\braket{\lambda,\underline{1}}}{n}}f(x)=e^{-\tfrac{\braket{\pi(x),\underline{1}}\braket{\lambda,\underline{1}}}{n}}f(\pi(x))=f(\pi(x))=G(\pi(\lambda),\pi(x)).$

The result follows by analytic extension. ∎

##  3\. Jack polynomials and Macdonald’s conjecture

We first recall some well-known facts about Jack polynomials from [KS97,
For10, Sta89]. Let
$\Lambda_{n}^{+}=\\{\lambda\in\mathbb{N}_{0}^{n}:\lambda_{1}\geq\ldots\geq\lambda_{n}\\}$
denote the set of partitions of length at most $n$. The dominance order on
$\Lambda_{n}^{+}$ is given by

$\mu\leq_{D}\lambda\,\,\text{ iff }\,\,|\lambda|=|\mu|\,\text{ and
}\sum_{j=1}^{r}\mu_{j}\leq\sum_{j=1}^{r}\lambda_{j}\text{ for all
}r=1,\ldots,n\,,$

where $|\lambda|=\lambda_{1}+\ldots+\lambda_{n}$. The dominance order is
extended from $\Lambda_{n}^{+}$ to $\mathbb{N}_{0}^{n}$ as follows: For each
composition $\eta\in\mathbb{N}_{0}^{n}$ denote by $\eta_{+}\in\Lambda_{n}^{+}$
the unique element in the $\mathcal{S}_{n}$-orbit of $\eta$. Then the
dominance order on $\mathbb{N}_{0}^{n}$ is defined by

$\kappa\preceq\eta\,\,\text{ iff
}\,\,\begin{cases}\kappa_{+}\leq_{D}\eta_{+}\,,&\kappa_{+}\neq\eta_{+}\\\
w_{\eta}\leq w_{\kappa}\,,&\kappa_{+}=\eta_{+}\end{cases},$

where $w_{\eta}\in\mathcal{S}_{n}$ is the shortest element with
$w_{\eta}\eta_{+}=\eta,$ and $\leq$ refers to the Bruhat order on
$\mathcal{S}_{n}$. Consider the rational Cherednik operators

$\mathcal{D}_{j}=\mathcal{D}_{j}(k):=x_{j}T_{j}+k(1-n)+k\sum_{i>j}s_{ij},\quad
1=1,\ldots,n$

where the $T_{j}:=T_{e_{j}}(k)$ are the type $A$ Dunkl operators with
multiplicity $k$ and $s_{ij}$ denotes the reflection in the root
$e_{i}-e_{j}$, which acts by interchanging $x_{i}$ and $x_{j}$. We remark that
our notion differs by a factor $k$ from that in [For10]. This facilitates the
handling of the case $k=0.$ The operators $\mathcal{D}_{j}$ are closely
related to the usual Cherednik operators $D_{j}:=D_{e_{j}}(k)$. Indeed,
consider $f\in C^{1}(U)$ for some open $U\subseteq\mathbb{R}^{n}$ and define
$g:\exp^{-1}(U)\subseteq\mathbb{R}^{n}\to\mathbb{C}$ by $g(x):=f(e^{x}),$
where $e^{x}$ is understood componentwise. Then a short calculation gives

$\bigl{(}D_{j}-\frac{k}{2}(n-1)\bigr{)}g(x)=(\mathcal{D}_{j}f)(e^{x}).$ (3.1)

The operators $\mathcal{D}_{j}$ are upper triangular with respect to $\preceq$
on $\mathcal{P}=\mathbb{C}[\mathbb{R}^{n}].$ More precisely,

$\mathcal{D}_{j}x^{\eta}\,=\,\overline{\eta}_{j}x^{\eta}+\sum_{\kappa\prec\eta}d_{\kappa\eta}x^{\kappa}$

with some $d_{\kappa\eta}\in\mathbb{R}$ and

$\overline{\eta}_{j}=\eta_{j}-k\\#\set{i<j\mid\eta_{i}\geq\eta_{j}}-k\\#\set{i>j\mid\eta_{i}>\eta_{j}}.$

The non-symmetric Jack polynomials of index $\alpha=1/k$ with $k\in[0,\infty)$
can be characterized as the unique basis
$\bigl{(}E_{\eta}=E_{\eta}(\,.\,;\alpha\bigr{)})_{\eta\in\mathbb{N}_{0}^{n}}$
of $\mathcal{P}$ satisfying

  1. (1)

$E_{\eta}(x)=x^{\eta}+\sum_{\kappa\prec\eta}c_{\eta\kappa}x^{\kappa}\,$ with
$\,c_{\kappa\eta}\in\mathbb{C}$,

  2. (2)

$\mathcal{D}_{j}E_{\eta}=\overline{\eta}_{j}E_{\eta}\,$ for all
$j=1,\ldots,n$.

By definition, $E_{\eta}$ is homogeneous of degree
$|\eta|=\eta_{1}+\ldots+\eta_{n}$, and for $k=0$ we have
$E_{\eta}(x;\infty)=x^{\eta}$. Property (2) together with Proposition 2.3 and
identity (3.1) show that the polynomials $E_{\eta}$ are related to the
(extended) Cherednik kernel via

$\frac{E_{\eta}\bigl{(}e^{x}\bigr{)}}{E_{\eta}(\underline{1})}\,=\,G\bigl{(}\overline{\eta}+\frac{k}{2}(n-1)\underline{1}\,,x\bigr{)},\quad\overline{\eta}=(\overline{\eta}_{1},\ldots,\overline{\eta}_{n}).$
(3.2)

Following [For10], we denote by
$P_{\lambda}(x)=P_{\lambda}(x;\alpha),\,\,\lambda\in\Lambda_{n}^{+}$ the
symmetric Jack polynomials in $n$ variables of index $\alpha=\frac{1}{k}$ in
monomial normalization. In the limiting case $k=0$, they coincide with the
monomial symmetric functions

$m_{\lambda}(x)=\sum_{\eta\in\mathcal{S}_{n}\lambda}x^{\eta}.$

The non-symmetric and symmetric Jack polynomials of the same index are related
via symmetrization: for $\lambda\in\Lambda_{n}^{+}$ and
$\eta\in\mathbb{N}_{0}^{n}$ with $\eta_{+}=\lambda,$

$\frac{P_{\lambda}(x)}{P_{\lambda}(\underline{1})}=\frac{1}{n!}\sum_{\sigma\in\mathcal{S}_{n}}\frac{E_{\eta}(\sigma
x)}{E_{\eta}(\underline{1})}.$ (3.3)

The Jack polynomials $P_{\lambda}$ satisfy a binomial formula:

$\frac{P_{\lambda}(\underline{1}+x)}{P_{\lambda}(\underline{1})}=\sum\limits_{\mu\subseteq\lambda}\binom{\lambda}{\mu}\frac{P_{\mu}(x)}{P_{\mu}(\underline{1})}\,,$
(3.4)

where $\mu\subseteq\lambda$ for $\lambda,\mu\in\Lambda_{n}^{+}$ means
$\mu_{i}\leq\eta_{i}$ for all $i=1,\ldots,n$ and
$\binom{\lambda}{\mu}=\binom{\lambda}{\mu}_{\\!k}\geq 0$ is a generalized
binomial coefficient.

Symmetrization in (3.2) yields a relation between the (extended)
hypergeometric function $F=F_{k}$ and the symmetric Jack polynomials: If
$\lambda\in\Lambda_{n}^{+}$, then

$\overline{\lambda}+\frac{k}{2}(n-1)\cdot\underline{1}\,=\,\lambda-\rho$ (3.5)

and therefore

$\frac{P_{\lambda}\bigl{(}e^{x}\bigr{)}}{P_{\lambda}(\underline{1})}\,=\,F(\lambda-\rho,x).$
(3.6)

In the following lemma, collect some further useful properties of the non-
symmetric Jack polynomials $E_{\eta}=E_{\eta}(\,.\,;\tfrac{1}{k})$, which can
be found in [For10, KS97, Sah98] for $k>0$ and are obvious for $k=0.$ Here we
consider the Jack polynomials as functions on $\mathbb{C}^{n}$.

######  Lemma 3.1.

  1. (1)

For all $p\in\mathbb{N}_{0}\,,$

$\Delta(z)^{p}E_{\eta}(z)=E_{\eta+\underline{p}}(z).$

By this property, the non-symmetric Jack polynomials uniquely extend to
indices $\eta\in\mathbb{Z}^{n}$.

  2. (2)

Let $z\in\mathbb{C}^{n}$ with $z_{i}\neq 0$ for all $\,i=1,\ldots,n.$ Then

$E_{\eta}\left(\tfrac{1}{z}\right)=E_{-\eta^{R}}(z^{R}),$

where $\eta^{R}=(\eta_{n},\ldots,\eta_{1}),\,z^{R}=(z_{n},\ldots,z_{1}).$

  3. (3)

Let $\Phi$ be the so-called raising operator, which acts on functions
$f:\mathbb{C}^{n}\to\mathbb{C}$ by

$\Phi f(z)=z_{n}f(z_{n},z_{1},\ldots,z_{n-1})$

and on $\mathbb{N}_{0}^{n}$ by

$\Phi\eta=(\eta_{2},\ldots,\eta_{n},\eta_{1}+1).$

Then the non-symmetric Jack polynomials satisfy

$\Phi E_{\eta}=E_{\Phi\eta}\quad(\eta\in\mathbb{N}_{0}^{n}).$

According to part (1) this identity extends to all $\eta\in\mathbb{Z}^{n}$,
because $\Phi(\Delta E_{\eta})=\Delta\cdot\Phi E_{\eta}$ and
$\Phi(\eta+\underline{p})=\Phi(\eta)+\underline{p}\,$ for all
$p\in\mathbb{N}.$

  4. (4)

The coefficients $c_{\eta\kappa}$ in the monomial expansion of
$E_{\eta},\eta\in\mathbb{N}_{0}^{n},$ are non-negative.

######  Lemma 3.2.

There exists a polynomial $Q\in\mathcal{P}$ such that for alle
$\eta\in\mathbb{N}_{0}^{n}$ and $\lambda\in\Lambda_{n}^{+},$

$0\leq E_{\eta}(\underline{1})\leq Q(\eta),\quad 0\leq
P_{\lambda}(\underline{1})\leq Q(\lambda).$

###### Proof.

By [For10, Prop. 12.3.2],

$E_{\eta}(\underline{1})=\prod_{(i,j)\in\eta}\frac{j+kn-k\ell^{\prime}(\eta,i,j)}{\eta_{i}-j+1+k\ell(\eta,i,j)+k}$

with the leg length and coleg length
$\ell(\eta,i,j),\ell^{\prime}(\eta,i,j)\in\\{0,\ldots,n\\}$. Therefore

$E_{\eta}(\underline{1})\leq\,\prod_{i=1}^{n}\prod_{j=1}^{\eta_{i}}\frac{j+kn}{\eta_{i}-j+1}=\prod_{i=1}^{n}\frac{(\eta_{i}+kn)!}{(kn)!\,\eta_{i}!}\,,$

which is polynomially bounded in $\eta$ by Stirling’s formula. Similarly (c.f.
[For10, Prop. 12.6.2]),

$P_{\lambda}(\underline{1})=\prod_{(i,j)\in\lambda}\frac{j-1+kn-
kl^{\prime}(\lambda,i,j)}{\lambda_{i}-j+kl(\eta,i,j)+k}\,\leq\prod_{i=1}^{n}\prod_{j=1}^{\lambda_{i}}\frac{j-1+kn}{\lambda_{i}-j+k}\,,$

which is also polynomially bounded in $\lambda$. ∎

To formulate the main results of this section, we introduce the gamma function

$\Gamma_{n}(\lambda)=\Gamma_{n}(k;\lambda):=d_{n}(k)\cdot\prod_{j=1}^{n}\Gamma(\lambda_{j}-k(j-1))\quad(\lambda\in\mathbb{C}^{n}),$
(3.7)

with

$d_{n}(k):=\prod_{j=1}^{n}\frac{\Gamma(1+jk)}{\Gamma(1+k)},$

as well as the generalized Pochhammer symbol

$[\mu]_{\eta}:=\prod_{j=1}^{n}(\mu-k(j-1))_{\eta_{j}}\,=\,\frac{\Gamma_{n}(\underline{\mu}+\eta)}{\Gamma_{n}(\underline{\mu})}\,\quad(\mu\in\mathbb{C},\eta\in\mathbb{N}_{0}^{n}).$

For abbreviation, we also write

$\Gamma_{n}(\mu):=\Gamma_{n}(\underline{\mu})\quad\text{ for
}\mu\in\mathbb{C}.$

Note that $\Gamma_{n}$ differs by the factor $d_{n}(k)$ from the notion in
[Rös20, Mac13], but is in accordance with the notion for the gamma function on
symmetric cones. We shall obtain the master theorem as a consequence of the
following result, which involves the type $A$ Dunkl operators $T=T(k)$ with
multiplicity $k$:

######  Theorem 3.3.

Fix $k\geq 0,$ and consider the non-symmetic Jack polynomials
$(E_{\eta})_{\eta\in\mathbb{N}_{0}^{n}}$ and the symmetric Jack polynomials
$(P_{\lambda})_{\lambda\in\Lambda_{n}^{+}}\,$ of index $1/k\,.$ Then for all
$\mu\in\mathbb{C}$ and all $x\in\mathbb{R}^{n}$ with $x_{i}\not=0$ for all
$i=1,\ldots,n$,

  1. (1)

$\displaystyle
E_{\eta}(T)\Delta^{-\mu}(x)=(-1)^{|\eta|}\,[\mu]_{\eta_{+}}E_{\eta}\left(\tfrac{1}{x}\right)\Delta(x)^{-\mu}$;

  2. (2)

$\displaystyle
P_{\lambda}(T)\Delta^{-\mu}(x)=(-1)^{|\lambda|}\,[\mu]_{\lambda}\,P_{\lambda}\left(\tfrac{1}{x}\right)\Delta(x)^{-\mu}$.

For the proof, we first note

######  Lemma 3.4.

The set $\mathbb{N}_{0}^{n}$ can be recursively constructed from
$0\in\mathbb{N}_{0}^{n}$ by a chain of the following operations:

  1. (i)

apply the raising operator $\Phi$ to $\eta\in\mathbb{N}_{0}^{n}$,

  2. (ii)

apply a simple permutation $s_{i}=(i,i+1)$ to $\eta\in\mathbb{N}_{0}^{n}$ with
$\eta_{i}<\eta_{i+1}$.

###### Proof.

This is easily verified by induction on the weight $|\eta|.$ Indeed, assume
that all elements of weight at most $r$ are already constructed and take
$\eta\in\mathbb{N}_{0}^{n}$ with $|\eta|=r+1$. Consider the maximal index
$j=1,\ldots,n$ with $\eta_{j}\neq 0$ and $\eta_{k}=0$ for $j<k\leq n$. Then

$\eta=(\eta_{1},\ldots,\eta_{j},0,\ldots,0)=s_{j}\cdots
s_{n-1}(\eta_{1},\ldots,\eta_{j-1},0,\ldots,0,\eta_{j})=s_{j}\cdots
s_{n-1}\Phi\widehat{\eta}$

with $\,\widehat{\eta}=(\eta_{j}-1,\eta_{1},\ldots,\eta_{j-1},0,\ldots,0),$
which is already constructed by induction hypothesis. ∎

###### Proof of Theorem 3.3.

Part (2) is obtained from (1) by symmetrization. Part (1) is clear for
$\eta=0,$ since $E_{0}=1.$ In view of the above observation, it therefore
suffices to consider the following two cases:

_Case 1._ Assume formula (1) is correct for some $\eta\in\mathbb{N}_{0}^{n}$
with $\eta_{i}<\eta_{i+1},$ and consider $E_{s_{i}\eta}.$ According to [For10,
Proposition 12.2.1] there exists a constant $d_{i}^{\eta}\in\mathbb{R}$ such
that

$E_{s_{i}\eta}=d_{i}^{\eta}E_{\eta}+s_{i}E_{\eta}.$

The Dunkl operators are $\mathcal{S}_{n}$-equivariant, i.e. $\sigma
T_{\xi}\sigma^{-1}=T_{\sigma\xi}$, $\sigma\in\mathcal{S}_{n}$. Hence the
symmetry of $\Delta(x)$ leads to

$\displaystyle(s_{i}E_{\eta})(T)\Delta(x)^{-\mu}$
$\displaystyle=(s_{i}E_{\eta}(T)(s_{i}\Delta)^{-\mu})(x)=E_{\eta}(T)\Delta^{-\mu}(s_{i}x)$
$\displaystyle=(-1)^{|\eta|}\,[\mu]_{\eta_{+}}E_{\eta}\bigl{(}\tfrac{1}{s_{i}x}\bigr{)}\Delta(s_{i}x)^{-\mu}$
$\displaystyle=(-1)^{|\eta|}\,[\mu]_{\eta_{+}}(s_{i}E_{\eta})\left(\tfrac{1}{x}\right)\Delta(x)^{-\mu}$

As $|s_{i}\eta|=|\eta|$ and $(s_{i}\eta)_{+}=\eta_{+},$ the formula follows
for $s_{i}\eta$ by linear combination.

_Case 2._ Assume that formula (1) is correct for some
$\eta\in\mathbb{N}_{0}^{n}$, and consider $\Phi\eta.$ Using the identity $\Phi
E_{\eta}=E_{\Phi\eta}$ from Lemma 3.1 and the product rule for the Dunkl
operators, we calculate

$\displaystyle E_{\Phi\eta}(T)\Delta(x)^{-\mu}$
$\displaystyle=T_{n}E_{\eta}(T_{n},T_{1},\ldots,T_{n-1})\Delta(x)^{-\mu}$
$\displaystyle=T_{n}\Big{(}(-1)^{|\eta|}\,[\mu]_{\eta_{+}}E_{\eta}(\tfrac{1}{x_{n}},\tfrac{1}{x_{1}},\ldots,\tfrac{1}{x_{n-1}})\Delta(x)^{-\mu}\Big{)}$
$\displaystyle=(-1)^{|\eta|}[\mu]_{\eta_{+}}\Big{(}(T_{n}\,\Delta(x)^{-\mu})E_{\eta}(\tfrac{1}{x_{n}},\tfrac{1}{x_{1}},\ldots,\tfrac{1}{x_{n-1}})$
$\displaystyle\quad+\Delta(x)^{-\mu}(T_{n}E_{\eta}(\tfrac{1}{x_{n}},\tfrac{1}{x_{1}},\ldots,\tfrac{1}{x_{n-1}}))\Big{)}.$
(3.8)

As $T_{n}$ acts on symmetric functions as the partial derivative
$\tfrac{\partial}{\partial x_{n}}$, we have
$T_{n}\Delta(x)^{-\mu}=-\mu\,x_{n}^{-1}\Delta(x)^{-\mu}$. Parts (1) and (2) of
Proposition 3.1 show that

$E_{\eta}(\tfrac{1}{x_{n}},\tfrac{1}{x_{1}},\ldots,\tfrac{1}{x_{n-1}})=\Delta^{-p}(x)E_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n})$

with $\,\eta^{*}=-\eta^{R}+\underline{p},$ where $p\in\mathbb{N}$ is so large
that $-\eta^{R}+\underline{p}\in\mathbb{N}_{0}^{n}$. Note further that
$\tfrac{1}{x_{n}}E_{\eta}(\tfrac{1}{x_{n}},\tfrac{1}{x_{1}},\ldots,\tfrac{1}{x_{n-1}})=E_{\Phi\eta}(\tfrac{1}{x}).$
Thus formula (3) reduces to

$\displaystyle E_{\Phi\eta}(T)\Delta(x)^{-\mu}$
$\displaystyle=(-1)^{|\eta|}[\mu]_{\eta_{+}}\Delta(x)^{-\mu}\big{(}-\mu
E_{\Phi\eta}(\tfrac{1}{x})+T_{n}(\Delta^{-p}(x)E_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n}))\big{)}.$
(3.9)

Again by the product rule for $T_{n}$ and that fact that $T_{n}$ commutes with
$s_{1},\ldots,s_{n-2},$ we further obtain

$\displaystyle T_{n}(\Delta^{-p}(x)E_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n}))$
$\displaystyle=-p\Delta(x)^{-p}E_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n})+\Delta(x)^{-p}(T_{n}E_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n}))$
$\displaystyle=-px_{n}^{-1}\Delta(x)^{-p}E_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n})+\Delta(x)^{-p}(T_{n}E_{\eta^{*}})(x_{n-1},\ldots,x_{1},x_{n})$
$\displaystyle=x_{n}^{-1}\Delta(x)^{-p}\Big{(}-pE_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n})+(x_{n}T_{n}E_{\eta^{*}})(x_{n-1},\ldots,x_{1},x_{n})\Big{)}.$

As $\,x_{n}T_{n}=\mathcal{D}_{n}+k(n-1),$ we have

$x_{n}T_{n}E_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n})=(\overline{\eta^{*}}_{n}+n-1)E_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n})$

with
$\,\overline{\eta^{*}}_{n}=\eta^{*}_{n}-k\\#\set{\ell<n\mid\eta^{*}_{\ell}\geq\eta_{n}^{*}}$,
so that

$\displaystyle T_{n}(\Delta^{-p}(x)E_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n}))$
$\displaystyle=\bigl{(}-p+(\overline{\eta^{*}}_{n}+k(n-1))\bigr{)}\Delta(x)^{-p}\tfrac{1}{x_{n}}E_{\eta^{*}}(x_{n-1},\ldots,x_{1},x_{n})$
$\displaystyle=\bigl{(}-p+(\overline{\eta^{*}}_{n}+k(n-1))\bigr{)}\tfrac{1}{x_{n}}E_{\eta}(\tfrac{1}{x_{n}},\tfrac{1}{x_{1}},\ldots,\tfrac{1}{x_{n-1}})$
$\displaystyle=\bigl{(}-p+(\overline{\eta^{*}}_{n}+k(n-1))\bigr{)}E_{\Phi\eta}(\tfrac{1}{x}).$

Thus equation (3) reduces to

$E_{\Phi\eta}(T)\Delta(x)^{-\mu}=(-1)^{|\eta|}[\mu]_{\eta_{+}}(-\mu-p+\overline{\eta^{*}}_{n}+k(n-1))E_{\Phi\eta}(\tfrac{1}{x})\Delta(x)^{-\mu}.$
(3.10)

Let $1\leq j\leq n$ be minimal such that the entry $j$ in $\eta_{+}$ is equal
to $\eta_{1}$, i.e.

$j-1=\\#\set{\ell>1\mid\eta_{1}<\eta_{\ell}}.$ (3.11)

Now at position $j$ in $(\Phi\eta)_{+}$ there is $\eta_{1}+1$. Thus, by
definition of $j$ we have

$\displaystyle\overline{\eta^{*}}_{n}$
$\displaystyle=(\eta^{R}+p)_{n}-k\\#\set{\ell<n\mid-\eta_{\ell}^{R}+p\geq-\eta_{n}^{R}+p}$
$\displaystyle=p+\eta_{1}-k\\#\set{\ell<n\mid\eta_{n-\ell+1}\leq\eta_{1}}$
$\displaystyle=p+\eta_{1}-k\\#\set{\ell>1\mid\eta_{\ell}\leq\eta_{1}}$
$\displaystyle=p+\eta_{1}-k(n-j).$

So finally, since $j$ is the position of $\eta_{1}+1=(\Phi\eta)_{n}$ in
$(\Phi\eta)_{+}$ we have that $(\Phi\eta)_{+}$ is exactly $\eta_{+}$ plus an
$1$ at position $j$. Therefore

$\displaystyle(-1)^{|\eta|}[\mu]_{\eta_{+}}(-\mu-p+\overline{\eta^{*}}_{n}+k(n-1))$
$\displaystyle=(-1)^{|\eta|}[\mu]_{\eta_{+}}(-\mu+\eta_{1}-k(n-j)+k(n-1))$
$\displaystyle=(-1)^{|\eta|+1}[\mu]_{\eta_{+}}(\mu-k(j-1)+(\eta_{1}+1)-1)$
$\displaystyle=(-1)^{|\Phi\eta|}[\mu]_{(\Phi\eta)_{+}}.$

Plugging this into (3.10) we obtain the assertion. ∎

######  Theorem 3.5 (Laplace transform identities for Jack polynomials).

Let $(E_{\eta})_{\eta\in\mathbb{N}_{0}^{n}}$ and
$(P_{\lambda})_{\lambda\in\Lambda_{n}^{+}}\,$ be the non-symmetric and
symmetric Jack polynomials of index $1/k\,,k\geq 0.$ Then for all
$\mu\in\mathbb{C}$ with $\text{Re}\,\mu>\mu_{0}$ and $z\in\mathbb{C}^{n}$ with
$\text{Re }z>0,$

  1. (1)

$\displaystyle\int_{\mathbb{R}_{+}^{n}}E(-z,x)E_{\eta}(x)\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx\,=\,\Gamma_{n}(\eta_{+}+\underline{\mu}\,)E_{\eta}\bigl{(}\tfrac{1}{z}\bigr{)}\Delta(z)^{-\mu};$

  2. (2)

$\displaystyle\int_{\mathbb{R}_{+}^{n}}J(-z,x)P_{\lambda}(x)\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx\,=\,\Gamma_{n}(\lambda+\underline{\mu}\,)P_{\lambda}\bigl{(}\tfrac{1}{z}\bigr{)}\Delta(z)^{-\mu}.$

Part (2) is just Macdonald’s [Mac13] Conjecture (C), and part (1) corresponds
to formula (4.38) in [BF98] (there is a misprint: the Laguerre polynomial
$E_{\eta}^{(L)}$ has to be replaced by $E_{\eta}\,).$

###### Proof.

The integrals converge by Lemma 2.1. According to [Rös20],

$\Delta(z)^{-\mu}=\,\frac{1}{\Gamma_{n}(\mu)}\mathcal{L}\bigl{(}\Delta^{\mu-\mu_{0}-1}\bigr{)}(z),$

and for each polynomial $p\in\mathcal{P}$,

$p(-T)\Delta^{-\mu}(z)=\,\frac{1}{\Gamma_{n}(\mu)}\mathcal{L}\bigl{(}p\,\Delta^{\mu-\mu_{0}-1}\bigr{)}(z).$

Now part (1) is immediate from Theorem 3.3 (1), and part (2) follows by
symmetrization. ∎

##  4\. Some general properties of the Cherednik kernel

We want to extend the Laplace transform identities for Jack polynomials to the
Cherednik kernel of type $A$. For this, we shall need some relations for the
Cherednik kernel which are of a general nature and seem not to be stated in
the literature. In this section, we therefore consider an arbitrary
crystallographic root system $R$ with Weyl group $W$ in a Euclidean space
$\mathfrak{a}$. In particular, it is required that $R$ spans $\mathfrak{a}.$
The inner product in $\mathfrak{a}$ is extended to $\mathfrak{a}_{\mathbb{C}}$
in a bilinear way. A $W$-invariant function $k:R\to\mathbb{C},\alpha\mapsto
k_{\alpha}$ is called a multiplicity function. The Cherednik operators
associated with multiplicity function $k$ and some positive subsystem $R_{+}$
of $R$ are defined by

$D_{\xi}(R_{+})=D_{\xi}(R_{+},k)=\partial_{\xi}-\langle\rho(R_{+}),\xi\rangle+\sum_{\alpha\in
R_{+}}k_{\alpha}\langle\alpha,\xi\rangle\frac{1-s_{\alpha}}{1-e^{-\alpha}}.$

where $\,\rho(R_{+})=\rho(R_{+},k)=\frac{1}{2}\sum_{\alpha\in
R_{+}}k_{\alpha}\alpha\,$ is the generalized Weyl vector. The
$D_{\xi}(R_{+}),$ $\xi\in\mathfrak{a}$ commute. According to [Opd95], there
exist a $W$-invariant tubular neighborhood $U$ of $\mathfrak{a}$ in
$\mathfrak{a}_{\mathbb{C}}$ such that for fixed multiplicity $k$ with
$\text{Re}\,k\geq 0\,$ and each $\lambda\in\mathfrak{a}_{\mathbb{C}}$ there
exists a unique holomorphic function
$\,f=G(\lambda,\cdot)=G_{k}(R_{+},\lambda,\cdot),$ called the Cherednik kernel
associated with $R_{+}$ and $k$, which satisfies the eigenvalue system

$\begin{cases}D_{\xi}(R_{+})f=\langle\lambda,\xi\,\rangle f\quad\text{for all
}x\in\mathfrak{a};\\\ f(0)=1.\end{cases}$

######  Proposition 4.1.

The Cherednik operators and the Cherednik kernel associated with the positive
system $R_{+}$ of $R$ have the following properties.

  1. (1)

$wD_{\xi}(R_{+})w^{-1}=D_{w\xi}(wR_{+})$ for all $w\in W$.

  2. (2)

$D_{\xi}(R_{+})f^{-}=-(D_{\xi}(R_{-})f)^{-},$ where $f^{-}(x)=f(-x)$ and
$\,R_{-}:=-R_{+}\,.$

  3. (3)

$G(R_{+},\lambda,z)=G(wR_{+},w\lambda,wz)$. In particular, the hypergeometric
function

$F(\lambda,z)=F_{k}(\lambda,z):=\frac{1}{|W|}\sum_{w\in W}G(R_{+},\lambda,wz)$

does not depend on the choice of $R_{+}$.

  4. (4)

$G(R_{+},\lambda,-z)=G(R_{+},-w_{0}\lambda,w_{0}z)$, where $w_{0}$ is the
longest element of $W.$

  5. (5)

$F(\lambda,-z)=F(-\lambda,z)$.

###### Proof.

(1) This follows from the identities

$w(\partial_{\xi}-\braket{\rho(R_{+}),\xi})w^{-1}=\partial_{w\xi}-\braket{\rho(wR_{+}),w\xi}$

and

$\displaystyle\sum\limits_{\alpha\in
R_{+}}k_{\alpha}\braket{\alpha,\xi}w\frac{1-s_{\alpha}}{1-e^{-\alpha}}w^{-1}$
$\displaystyle=\sum\limits_{\beta\in
wR_{+}}k_{\beta}\braket{\beta,\xi}\frac{1-s_{\beta}}{1-e^{-\beta}}.$

(2) Note that
$(\partial_{\xi}-\braket{\rho(R_{+}),\xi})f^{-}=-(\partial_{\xi}-\braket{\rho(-R_{+}),\xi}f)^{-}$.
As $k_{\alpha}=k_{-\alpha},$ we also have

$\displaystyle\sum\limits_{\alpha\in
R_{+}}k_{\alpha}\braket{\alpha,\xi}\frac{f^{-}-s_{\alpha}f^{-}}{1-e^{-\alpha}}$
$\displaystyle=-\left(\sum\limits_{\beta\in-
R_{+}}k_{\beta}\braket{\beta,\xi}\frac{1-s_{\beta}}{1-e^{-\beta}}f\right)^{-}.$

Thus $\,D_{\xi}(R_{+})f^{-}=-(D_{\xi}(-R_{+})f)^{-}$.

(3) In view of part (1), the defining eigenvalue equation
$D_{\xi}(R_{+})f=\braket{\lambda,\xi}f$ is equivalent to

$D_{w\xi}(wR_{+})(wf)=wD_{\xi}(R_{+})f=\braket{\lambda,\xi}(wf)=\braket{w\lambda,w\xi}(wf).$

Hence $G(R_{+},\lambda,w^{-1}z)=G(wR_{+},w\lambda,z)$.

(4) From part (2) we observe that

$D_{\xi}(R_{-})G(\lambda,-\cdot)=-(D_{\xi}(R_{+})G_{k}(\lambda,\cdot))^{-}=-\braket{\lambda,\xi}G(\lambda,-\cdot).$

Therefore $G(R_{+},\lambda,-z)=G(R_{-},-\lambda,z)$. The longest element
$w_{0}\in W$ satisfies $\,w_{0}R_{-}=R_{+}$. Hence by part (3),

$G(R_{+},\lambda,-z)=G(w_{0}R_{-},-w_{0}\lambda,w_{0}z)=G(R_{+},-w_{0}\lambda,w_{0}z).$

(5) This is clear from part (4). ∎

If $k\geq 0,$ then according to [Sch08] and [Opd95], $G(\lambda,x)>0$ for all
$\lambda,x\in\mathfrak{a}$ and

$|G(\lambda,x)|\leq G(\text{Re}\,\lambda,x)\leq\sqrt{|W|}\,e^{\max_{w\in
W}\langle\text{Re}\,\lambda,wx\rangle}\quad\text{ for all
}\,\lambda\in\mathfrak{a}_{\mathbb{C}},\,x\in\mathfrak{a}.$ (4.1)

The following result generalizes the estimate of the Cherednik kernel stated
in [RKV13, Theorem 3.3]. Notice that the eigenvalue characterization of $G$
implies that $G(-\rho,\cdot)\equiv 1$ for $\rho=\rho(R_{+}).$

######  Proposition 4.2.

Suppose that $k\geq 0.$ Then the Cherednik kernel $G=G_{k}(R_{+},\cdot,\cdot)$
satisfies the following estimate for $x\in\mathfrak{a}$ and all
$\lambda,\mu\in\mathfrak{a}_{\mathbb{C}}$:

$\left|G(\lambda+\mu,x)\right|\leq G(\text{Re}\,\mu\,,x)\cdot e^{\max_{w\in
W}\braket{\text{Re}\,\lambda,wx}}.$

Moreover, since $G(-\rho,\cdot)\equiv 1,$ we in particular have

$\left|G(\lambda-\rho,x)\right|\leq e^{\max_{w\in
W}\braket{\text{Re}\,\lambda,wx}}\quad\textrm{ for all
}\lambda\in\mathfrak{a}_{\mathbb{C}},\,x\in\mathfrak{a}.$

###### Proof.

In [RKV13, Theorem 3.3] it was proven by a Phragmen-Lindelöf argument that for
all $\lambda\in\mathfrak{a},\,\mu\in\overline{\mathfrak{a}_{+}}\,$ and
$x\in\mathfrak{a}$,

$G(\lambda+\mu,x)\,\leq\,G(\mu,x)\cdot e^{\max_{w\in W}\braket{\lambda,wx}}.$
(4.2)

An inspection of the proof in loc.cit. shows that it can be carried out in
exactly the same way when $\overline{\mathfrak{a}_{+}}$ is replaced by an
arbitrary closed Weyl chamber $C\subset\mathfrak{a}.$ Therefore estimate (4.2)
extends to all $\mu\in\mathfrak{a}$, and the claim follows from (4.1). ∎

######  Remark 4.3.

For root system $R=A_{n-1}$, items (3)–(5) of Proposition 4.1 as well as
Proposition 4.2 are easily checked to remain valid for the extensions of $G$
and $F$ as defined in (2.2).

##  5\. Laplace transform of Heckman-Opdam hpergeometric functions and the
Cherednik kernel

In this section, we return to $A_{n-1}\,$ and resume the notations from
Sections 1–3. We shall extend the statements of Theorem 3.5 to the Cherednik
kernel and the hypergeometric function of type $A_{n-1}.$ Formulas (3.2) and
(3.6) suggest that it will be convenient to work with suitable modifications
of the (extended) kernels $G$ and $F.$ According to our construction in
Section 2, the extended Cherednik kernel $G$ is in particular holomorphic on
the set

$\mathbb{C}^{n}\times(\mathbb{R}^{n}+i\Omega^{\prime})\,\text{ with
}\,\Omega^{\prime}=\\{x\in\mathbb{R}^{n}:\,|x_{i}|<\tfrac{\pi}{2}\,\text{ for
all }\,i=1,\ldots,n\\}.$

The exponential mapping $\exp:z\mapsto e^{z}$ (understood componentwise) maps
$\mathbb{R}^{n}+i\Omega^{\prime}$ biholomorphically onto
$H:=\\{z\in\mathbb{C}^{n}:\text{Re}\,z>0\\}$ with inverse $\log$. The modified
kernels

$\displaystyle\mathcal{G}(\lambda,z):=G(\lambda,\log(z)),$
$\displaystyle\mathcal{F}(\lambda,z):=F(\lambda,\log(z))$

are therefore holomorphic on $\mathbb{C}^{n}\times H.$ We call them the
rational version of the Cherednik kernel and the hypergeometric function,
respectively. Obviously $\mathcal{F}$ is $\mathcal{S}_{n}$-invariant in each
argument. We collect some properties of $\mathcal{G}$ and $\mathcal{F}$ which
shall be needed lateron.

######  Lemma 5.1.

  1. (1)

For all $z\in H,\lambda\in\mathbb{C}^{n}$ and $\mu\in\mathbb{C}$,

$\Delta(z)^{\mu}\,\mathcal{G}(\lambda,z)=\mathcal{G}(\lambda+\underline{\mu},z),\quad\Delta(z)^{\mu}\,\mathcal{F}(\lambda,z)=\mathcal{F}(\lambda+\underline{\mu},z).$

  2. (2)

For all $z\in H$ and $\lambda\in\mathbb{C}^{n}$,

$\mathcal{G}(\lambda,\tfrac{1}{z})=\mathcal{G}(-\lambda^{R},z^{R}),\quad\mathcal{F}(\lambda,\tfrac{1}{z})=\mathcal{F}(-\lambda,z),$

  3. (3)

For all $x\in\mathbb{R}_{+}^{n}$ and all $\lambda,\mu\in\mathbb{C}^{n}$,

$|\mathcal{G}(\lambda+\mu,x)|\,\leq\,\mathcal{G}(\text{Re}\,\lambda,x)\cdot\max_{\sigma\in\mathcal{S}_{n}}x^{\sigma(\text{Re}\,\mu)}.$

In particular, for all $x\in\mathbb{R}_{+}^{n}$ and
$\lambda\in\mathbb{C}^{n}$,

$|\mathcal{G}(\lambda,x)|\,\leq\,\max_{\sigma\in\mathcal{S}_{n}}x^{\sigma(\text{Re}\,\lambda)}.$

The same estimates hold for $\mathcal{F}.$

  4. (4)

For partitions $\lambda\in\Lambda_{n}^{+}$,

$\mathcal{G}(\lambda-\rho,z)=\frac{E_{\lambda}(z)}{E_{\lambda}(\underline{1})}\,.$

###### Proof.

Part (1) is clear from the definitions. Part (2) follows from Proposition 4.1,
since the longest element $w_{0}\in\mathcal{S}_{n}$ acts by
$w_{0}\lambda=\lambda^{R}=(\lambda_{n},\ldots,\lambda_{1}).$ Part (3) is
immediate from Proposition 4.2 (and the subsequent remark). Finally, part (4)
follows from identities (3.2) and (3.5). ∎

The extension of Theorem3.5 will be carried out by analytic extension with
respect to the spectral parameter, which is based on the following
generalization of the classical Carlson theorem [Tit76, p.186].

######  Lemma 5.2.

Let $U\subseteq\mathbb{C}^{n}$ be an open neighborhood of $\\{\text{Re}\,z\geq
0\\}\subseteq\mathbb{C}^{n}$ and let $f:U\to\mathbb{C}$ be holomorphic. Put
$\left\lVert z\right\rVert_{1}:=\sum_{i=1}^{n}\left|z_{i}\right|$. If $f$
satisfies

$(\ast)\quad f(z)=\mathcal{O}(e^{c\left\lVert z\right\rVert_{1}})\,\textrm{
for some }c<\pi\textrm{ and }\,f|_{\Lambda_{n}^{+}}\equiv 0,$

then $f\equiv 0$.

###### Proof.

We proceed by induction on $n$. The case $n=1$ is Carlson’s classical theorem.
To achieve step $n-1\to n$, consider for fixed $\lambda\in\Lambda_{n}^{+}$ the
holomorphic function

$f_{\lambda}:U^{\prime}\to\mathbb{C},\xi\mapsto
f(\xi+\lambda_{1},\lambda_{2},\ldots,\lambda_{n})$

where $U^{\prime}\subseteq\mathbb{C}$ is a suitable neighborhood of
$\\{\text{Re}\,\xi\,+\lambda_{1}\geq 0\\}\subseteq\mathbb{C}$. Then
$f_{\lambda}|_{\mathbb{N}_{0}}\equiv 0$ and

$f_{\lambda}(\xi)=\mathcal{O}(e^{c\left|\xi\right|})$

with $c$ as in $(\ast).$ Therefore $f_{\lambda}$ vanishes identically. From
this we conclude that for fixed $\xi\in\mathbb{C}$ with $\text{Re}\,\xi\geq
0$, the function

$g_{\xi}:\widetilde{U}\to\mathbb{C},\quad w\mapsto f(\xi,w)$

vanishes on $\Lambda_{n-1}^{+}$ for some suitable neighborhood
$\widetilde{U}\subseteq\mathbb{C}^{n-1}$ of $\\{\text{Re}\,w\geq 0\\}$.
Moreover

$g_{\xi}(w)=\mathcal{O}(e^{c\left\lVert(\xi,w)\right\rVert_{1}})=\mathcal{O}(e^{c\left\lVert
w\right\rVert_{1}}),$

and by the induction hypothesis we obtain that $g_{\xi}$ vanishes identically.
As $\xi$ was arbitrary, we obtain $f\equiv 0$. ∎

We are now in the position to prove the following generalization of Theorem
3.5:

######  Theorem 5.3.

Let $\mu\in\mathbb{C}$ with
$\text{Re}\,\mu>\mu_{0}=k(n-1),\,\lambda\in\mathbb{C}^{n}$ with
$\text{Re}\,\lambda\geq 0$ and $z\in H.$ Then

  1. (1)

$\displaystyle\int_{\mathbb{R}_{+}^{n}}E(-z,x)\,\mathcal{G}(\lambda,x)\,\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx\,=\,\Gamma_{n}(\lambda+\rho+\underline{\mu})\,\mathcal{G}(\lambda,\tfrac{1}{z})\,\Delta(z)^{-\mu}.$

  2. (2)

$\displaystyle\int_{\mathbb{R}_{+}^{n}}J(-z,x)\,\mathcal{F}(\lambda,x)\,\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx\,=\,\Gamma_{n}(\lambda+\rho+\underline{\mu})\,\mathcal{F}(\lambda,\tfrac{1}{z})\,\Delta(z)^{-\mu}.$

The second formula generalizes the Laplace transform identity (1.5) for
spherical functions on a symmetric cone. In view of Lemma 5.1, this theorem
can be equivalently reformulated as follows:

######  Corollary 5.4.

Suppose that $\text{Re}\,\lambda\geq\underline{\mu_{0}}\,$. Then for all $z\in
H,$

  1. (1)

$\displaystyle\int_{\mathbb{R}_{+}^{n}}E(-z,x)\,\mathcal{G}(\lambda,x)\,\Delta(x)^{-\mu_{0}-1}\omega(x)dx\,=\,\Gamma_{n}(\lambda+\rho)\,\mathcal{G}(\lambda,\tfrac{1}{z});$

  2. (2)

$\displaystyle\int_{\mathbb{R}_{+}^{n}}J(-z,x)\,\mathcal{F}(\lambda,x)\,\Delta(x)^{-\mu_{0}-1}\omega(x)dx\,=\Gamma_{n}(\lambda+\rho)\,\mathcal{F}(\lambda,\tfrac{1}{z}).$

###### Proof of Theorem 5.3.

It suffices to check part (1). By Carlson’s theorem, we shall prove that

$\frac{1}{\Gamma_{n}(\lambda+\rho+\underline{\mu})}\int_{\mathbb{R}_{+}^{n}}E(-z,x)\,\mathcal{G}(\lambda,x)\,\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx\,=\,\mathcal{G}(\lambda,\tfrac{1}{z})\,\Delta(z)^{-\mu}.$
(5.1)

Note first that (5.1) holds for all $\,\lambda\in\Lambda_{n}^{+}-\rho\,,$ by
Theorem 3.5 and Lemma 5.1(4). The right hand side of (5.1) is holomorphic in
$(\lambda,z,\mu)$ on $\mathbb{C}^{n}\times H\times\mathbb{C}.$ Moreover, the
left hand side exists and is continuous on $\overline{H}\times
H\times\\{\text{Re}\,\mu>\mu_{0}\\}$ and holomorphic on $H\times
H\times\\{\text{Re}\,\mu>\mu_{0}\\}$. Indeed, suppose that
$\text{Re}\,z\geq\underline{s}$ for some $s>0.$ Then by [Rös20],

$\left|E(-z,x)\right|\leq
E(-\text{Re}\,z,x)\,\leq\,e^{-\langle\underline{s},x\rangle}.$

Together with Lemma 5.1 we obtain for $x\in\mathbb{R}_{+}^{n}$

$\big{|}E(-z,x)\,\mathcal{G}(\lambda,x)\Delta(x)^{\mu-\mu_{0}-1}\big{|}\,\leq\,e^{-\langle\underline{s},x\rangle}\Delta(x)^{\text{Re}\,\mu-\mu_{0}-1}\max_{\sigma\in\mathcal{S}_{n}}x^{\sigma(\text{Re}\,\lambda)}.$

Hence the integral on the left hand side of formula (5.1) exists and is (by
standard arguments) continuous respectively holomorphic as stated. It
therefore suffices to check (5.1) for $z\in\mathbb{R}^{n}$ with
$z>\underline{1}$ and $\mu\in\mathbb{R}$ with $\mu>\mu_{0}$. We want to apply
Carlson’s Theorem 5.2 with respect to $\lambda$. As $z>\underline{1},$ the
right hand side of (5.1) is bounded in $\lambda$ according to Lemma 5.1, and
it remains to control the growth of the left hand side. For $\lambda\in H,$
define
$\,\eta(\lambda):=(\lceil\text{Re}\,\lambda_{1})\rceil,\ldots,\lceil\text{Re}\,\lambda_{n})\rceil)_{+}\in\Lambda_{n}^{+}\,.$
Then for arbitrary $x\in\mathbb{R}_{+}^{n},$

$\max\limits_{\sigma\in\mathcal{S}_{n}}x^{\sigma(\text{Re}\,\lambda)}\,\leq\,\max\limits_{\sigma\in\mathcal{S}_{n}}(\underline{1}+x)^{\sigma(\text{Re}\,\lambda)}\leq\,P_{\eta(\lambda)}(\underline{1}+x),$

because the coefficients of $P_{\eta(\lambda)}$ in its monomial expansion are
nonnegative. Now recall the binomial formula (3.4) for the Jack polynomials as
well as the identity

$\int_{\mathbb{R}_{+}^{n}}e^{-\langle\underline{1},x\rangle}P_{\kappa}(x)\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx\,=\,\Gamma_{n}(\kappa+\underline{\mu})P_{\kappa}(\underline{1})$

from [Mac13, (6.18)] (c.f. also [Rös20, Lemma 5.1]). We may therefore estimate

$\displaystyle\int_{\mathbb{R}_{+}^{n}}$
$\displaystyle\left|E(-z,x)\,\mathcal{G}(\lambda,x)\,\Delta(x)^{\mu-\mu_{0}-1}\right|\omega(x)dx$
$\displaystyle\leq\int_{\mathbb{R}_{+}^{n}}e^{-\langle\underline{1},x\rangle}\,P_{\eta(\lambda)}(\underline{1}+x)\,\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx$
$\displaystyle=\,\sum\limits_{\kappa\subseteq\eta(\lambda)}\binom{\eta(\lambda)}{\kappa}\int_{\mathbb{R}_{+}^{n}}e^{-\langle\underline{1},x\rangle}P_{\kappa}(x)\,\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx$
$\displaystyle=\sum\limits_{\kappa\subseteq\eta(\lambda)}\binom{\eta(\lambda)}{\kappa}\,P_{\kappa}(\underline{1})\,\Gamma_{n}(\kappa+\underline{\mu}).$

By monotonicity of the classical gamma function,

$\Gamma_{n}(\kappa+\underline{\mu})\leq\Gamma_{n}(\eta(\lambda)+\underline{\mu})\leq\Gamma_{n}((\text{Re}\,\lambda)_{+}+\underline{1}+\underline{\mu}).$

Moreover, by Remark 3.2,

$\sum\limits_{\kappa\subseteq\eta(\lambda)}\binom{\eta(\lambda)}{\kappa}P_{\kappa}(\underline{1})\,=\,P_{\eta(\lambda)}(\underline{2})\,=\,2^{|\eta(\lambda)|}P_{\eta(\lambda)}(\underline{1})\,\leq\,2^{\|\lambda\|_{1}}\cdot
Q(\lambda)$

with some polynomial $Q\in\mathcal{P}.$ Therefore

$\displaystyle I_{z,\mu}(\lambda)$
$\displaystyle:=\,\left|\frac{1}{\Gamma_{n}(\lambda+\rho+\underline{\mu})}\int_{\mathbb{R}_{+}^{n}}E(-z,x)\,\mathcal{G}(\lambda,x)\,\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx\right|$
$\displaystyle\,\leq\,Q(\lambda)\cdot\frac{\Gamma_{n}((\text{Re}\,\lambda)_{+}+1+\underline{\mu})}{\left|\Gamma_{n}(\lambda+\rho+\underline{\mu})\right|}\,\cdot
2^{\|\lambda\|_{1}}.$

Choose $\sigma\in\mathcal{S}_{n}$ with
$\,\sigma(\text{Re}\,\lambda)=(\text{Re}\,\lambda)_{+}$ . Then

$\displaystyle\frac{\Gamma_{n}((\text{Re}\,\lambda)_{+}+1+\underline{\mu})}{\left|\Gamma_{n}(\lambda+\rho+\underline{\mu})\right|}=\prod\limits_{j=1}^{n}\frac{\Gamma\bigl{(}((\text{Re}\,\lambda)_{+})_{j}+\mu+1-k(j-1)\bigr{)}}{\left|\Gamma\bigl{(}\lambda_{j}+\mu+\rho(k)_{j}-k(j-1)\bigr{)}\right|}\,=\,F_{1}(\lambda)\cdot
F_{2}(\lambda)$

with

$\displaystyle F_{1}(\lambda)$
$\displaystyle=\,\prod_{j=1}^{n}\frac{\Gamma\bigl{(}\text{Re}\,\sigma(\lambda)_{j}+\mu+1-k(j-1)\bigr{)}}{\Gamma\bigl{(}\text{Re}\,\sigma(\lambda)_{j}+\mu+1-\frac{k}{2}(n-1)\bigr{)}}\,,$
$\displaystyle F_{2}(\lambda)$
$\displaystyle=\,\prod_{j=1}^{n}\frac{\left|\lambda_{j}+\mu-\frac{k}{2}(n-1)\right|\cdot\Gamma\bigl{(}\text{Re}\,\lambda_{j}+\mu+1-\frac{k}{2}(n-1)\bigr{)}}{\left|\Gamma\bigl{(}\lambda_{j}+\mu+1-\frac{k}{2}(n-1)\bigr{)}\right|}\,.$

By Stirling’s formula, $F_{1}(\lambda)$ is polynomially bounded, i.e.
$F_{1}(\lambda)=\mathcal{O}(e^{\epsilon\|\lambda\|_{1}})$ for arbitrary
$\epsilon>0.$ For $F_{2}$, we employ the estimate ([N10, Formula 5.6.7])

$\frac{\Gamma(x)}{\left|\Gamma(x+iy)\right|}\leq\sqrt{\cosh(\pi
y)}=\mathcal{O}(e^{\tfrac{\pi}{2}|y|}),\quad x>\tfrac{1}{2},y\in\mathbb{R},$

which leads to

$F_{2}(\lambda)\,=\,\mathcal{O}(e^{(\epsilon+\tfrac{\pi}{2})\|\lambda\|_{1}})$

with arbitrary $\epsilon>0.$ Putting things together, we obtain that
$I_{z,\mu}(\lambda)$ satisfies the growth condition of Carlson’s Theorem 5.2,
which finishes the proof. ∎

##  6\. Macdonald’s hypergeometric series and their Laplace transform

In the setting of symmetric cones, the Laplace transform establishes important
identities between hypergeometric series. Analogous formulas were formally
stated by Macdonald [Mac13] for general Jack-hypergeometric series, as
consequences of his conjecture $(C).$ With Theorem 3.5 at hand, we shall make
these identities precise, and extend them to hypergeometric expansions in
terms of non-symmetric Jack polynomials. We start with the appropriate
normalization of the symmetric and non-symmetric Jack polynomials.

######  Lemma 6.1.

  1. (i)

There are numbers $c_{\eta}>0$ for $\eta\in\mathbb{N}_{0}^{n}$ such that the
renormalized Jack polynomials $\,C_{\lambda}:=c_{\lambda}P_{\lambda}\,$ and
$\,L_{\eta}:=c_{\eta}E_{\eta}\,$ satisfy

$\sum_{\lambda\in\Lambda_{n}^{+}:|\lambda|=m}C_{\lambda}(z)=\sum_{\eta\in\mathbb{N}_{0}^{n}:|\eta|=m}L_{\eta}(z)\,=\,(z_{1}+\ldots+z_{n})^{m}\quad\text{for
all }m\in\mathbb{N}_{0};$

  2. (ii)

$\displaystyle
C_{\lambda}=\sum\limits_{\eta\in\mathcal{S}_{n}\lambda}L_{\eta}\,\,$ for all
$\,\lambda\in\Lambda_{n}^{+}$.

  3. (iii)

$\displaystyle c_{\lambda}\leq\,\frac{|\lambda|!}{\lambda!}\,$ for all
$\,\lambda\in\Lambda_{n}^{+}$.

###### Proof.

We may assume that $k>0.$ Part (i) for the symmetric Jack polynomials is well-
known (see e.g. [For10, (12.135)]), with

$c_{\lambda}=\frac{|\lambda|!}{k^{|\lambda|}d_{\lambda}^{\prime}}.$

Here the constants $d_{\eta}^{\prime}$ for $\eta\in\mathbb{N}_{0}^{n}$ are
given by

$d_{\eta}^{\prime}=\prod_{(i,j)\in\eta}\Bigl{(}\frac{1}{k}(\eta_{i}-j+1)+\ell(\eta,i,j)\Bigr{)}>0,$

with the leg length $\,\ell(\eta,i,j)=\\#\set{\ell>i\mid
j\leq\eta_{\ell}\leq\eta_{i}}\,+\,\\#\set{\ell<i\mid
j\leq\eta_{\ell}+1\leq\eta_{i}}$. In particular, for each partition
$\lambda\in\Lambda_{n}^{+}$ we have

$c_{\lambda}=\frac{\left|\lambda\right|!}{\prod\limits_{(i,j)\in\lambda}((\lambda_{i}-j+1)+k\ell(\lambda,i,j))}\leq\frac{\left|\lambda\right|!}{\prod\limits_{1\leq
j\leq\lambda_{i}}(\lambda_{i}-j+1)}=\frac{\left|\lambda\right|!}{\lambda!},$

which is part (iii). From [For10, Proposition 12.6.1] it is further known that

$P_{\lambda}=d_{\lambda}^{\prime}\sum\limits_{\eta\in\mathcal{S}_{n}\lambda}\frac{1}{d_{\eta}^{\prime}}E_{\eta}.$
(6.1)

Hence, we put

$c_{\eta}:=c_{\eta_{+}}\frac{d_{\eta_{+}}^{\prime}}{d_{\eta}^{\prime}}=\frac{|\eta|!}{k^{|\eta|}\,d_{\eta}^{\prime}}$

for $\eta\in\mathbb{N}_{0}^{n}$, and part (i) for the non-symmetric Jack
polynomials follows. Finally, part (ii) is immediate from the definition of
$c_{\eta}$ and relation (6.1).

∎

On the space $\mathcal{P}_{\mathbb{R}}=\mathbb{R}[\mathbb{R}^{n}]$ of real
polynomials on $\mathbb{R}^{n}$ there exists an $\mathcal{S}_{n}$-invariant
inner product $[\cdot,\cdot]=[\cdot,\cdot]_{k}$ called the Dunkl pairing (cf.
[Dun91]), which is defined by

$[p,q]:=(p(T)q)(0).$

Here the Dunkl operators are again those of type $A_{n-1}$ with multiplicity
$k.$ Polynomials with different homogeneous degree are orthogonal with respect
to this pairing, and $\,[T_{\xi}p,q]=[p,\braket{\cdot,\xi}q].$ This property
and the invariance under the action of $\mathcal{S}_{n}$ show that the
Cherednik operators $\mathcal{D}_{j}$ are symmetric with respect to the Dunkl
pairing. In particular, the non-symmetric Jack polynomials
$(E_{\eta})_{\eta\in\mathbb{N}_{0}^{n}}$ form an orthogonal basis of
$\mathcal{P}_{\mathbb{R}}$ with respect to $[\,.\,,\,.].$ More precisely,
their renormalizations $L_{\eta}=c_{\eta}E_{\eta}$ satisfy

$[L_{\eta},L_{\kappa}]=\,|\eta|!\,L_{\eta}(\underline{1})\cdot\delta_{\eta,\kappa}$
(6.2)

which is obtained by combining Propos. 3.18 and formula (2.4) of [BF98].

######  Lemma 6.2.

The Dunkl kernel of type $A_{n-1}$ with multiplicity $k\geq 0$ satisfies

$E(z,w)=\sum_{\eta\in\mathbb{N}_{0}^{n}}\frac{L_{\eta}(z)L_{\eta}(w)}{|\eta|!\,L_{\eta}(\underline{1})}.$

The series converges locally uniformly on
$\mathbb{C}^{n}\times\mathbb{C}^{n}.$

###### Proof.

This is immediate from [Rös98, Lemma 3.1] together with identity (6.2).
Alternatively, the stated expansion follows from [For10, Propos. 13.3.4]. ∎

######  Definition 6.3.

Consider the Jack polynomials $(L_{\eta})_{\eta\in\mathbb{N}_{0}^{n}}$ and
$(C_{\lambda})_{\lambda\in\Lambda_{n}^{+}}$ of index $\alpha=\tfrac{1}{k}$,
respectively (normalized as above). Following [Mac13], [Kan93] and [BF98], we
define for indices $\mu\in\mathbb{C}^{p}$ and $\nu\in\mathbb{C}^{q}$ with
$p,q\in\mathbb{N}_{0}$ the non-symmetric hypergeometric series

${}_{p}K_{q}(\mu;\nu;z,w)\,:=\sum\limits_{\eta\in\mathbb{N}_{0}^{n}}\frac{[\mu_{1}]_{\eta_{+}}\cdots[\mu_{p}]_{\eta_{+}}}{[\nu_{1}]_{\eta_{+}}\cdots[\nu_{q}]_{\eta_{+}}}\,\frac{L_{\eta}(z)L_{\eta}(w)}{\left|\eta\right|!\,L_{\eta}(\underline{1})}$

as well as the symmetric hypergeometric series

${}_{p}F_{q}(\mu;\nu;z,w)\,:=\sum\limits_{\lambda\in\Lambda_{n}^{+}}\frac{[\mu_{1}]_{\lambda}\cdots[\mu_{p}]_{\lambda}}{[\nu_{1}]_{\lambda}\cdots[\nu_{q}]_{\lambda}}\,\frac{C_{\lambda}(z)C_{\lambda}(w)}{\left|\eta\right|!\,C_{\lambda}(\underline{1})}.$

More common in the literature are hypergeometric series in one variable, which
are obtained as functions in $z$ by setting $w=\underline{1}$.
For abbreviation, we write for $\lambda\in\Lambda_{n}^{+}$

$[\mu]_{\lambda}:=[\mu_{1}]_{\lambda}\cdots[\mu_{p}]_{\lambda};\quad[\nu]_{\lambda}:=[\nu_{1}]_{\lambda}\cdots[\nu_{q}]_{\lambda}.$

Note that for $p=0$ or $q=0,$ an empty product occurs. For those values of $k$
for which the $C_{\lambda}=C_{\lambda}(\cdot\,;\frac{1}{k})$ are the spherical
polynomials of a symmetric cone, the convergence properties of
$\,{}_{p}F_{q}$-hypergeometric series in one variable are well-known, see
[FK94, GR89]. For general $k>0,$ partial results on the domain of convergence
of $\,{}_{p}F_{q}$ were obtained in [Kan93]. For some values of $p$ and $q$,
the nonsymmetric series ${}_{p}K_{q}$ were considered in [BF98]. But to our
knowledge, their convergence properties have not been studied so far.

######  Lemma 6.4.

The non-symmetric and symmetric hypergeometric functions are related by

$\frac{1}{n!}\sum\limits_{\sigma\in\mathcal{S}_{n}}{}_{p}K_{q}(\mu;\nu;\sigma
z,w)={}_{p}F_{q}(\mu;\nu;z,w).$

###### Proof.

By identity (3.3) and Lemma 6.1 we have

$\displaystyle\frac{1}{n!}\sum\limits_{\sigma\in\mathcal{S}_{n}}{}_{p}K_{q}(\mu;\nu;\sigma
z,w)$
$\displaystyle=\sum\limits_{\eta\in\mathbb{N}_{0}^{n}}\frac{[\mu]_{\eta_{+}}}{[\nu]_{\eta_{+}}}\frac{1}{\left|\eta\right|!}L_{\eta}(w)\frac{C_{\eta_{+}}(z)}{C_{\eta_{+}}(\underline{1})}$
$\displaystyle=\sum\limits_{\lambda\in\Lambda_{n}^{+}}\frac{[\mu]_{\lambda}}{[\nu]_{\lambda}}\frac{1}{\left|\lambda\right|!}\Bigl{(}\sum\limits_{\eta\in\mathcal{S}_{n}\lambda}L_{\eta}(w)\Bigr{)}\,\frac{C_{\lambda}(z)}{C_{\lambda}(\underline{1})}$
$\displaystyle={}_{p}F_{q}(\mu;\nu;w,w).$

∎

######  Theorem 6.5.

Let $\mu\in\mathbb{C}^{p}$ and $\nu\in\mathbb{C}^{q}$ with
$\nu_{i}\notin\set{0,k,\ldots,k(n-1)}-\mathbb{N}_{0}$ for all $i=1,\ldots,n$
(i.e. $[\nu]_{\lambda}\neq 0$ for all $\lambda\in\Lambda_{n}^{+})$.

  1. (1)

If $p\leq q$, the series ${}_{p}K_{q}(\mu;\nu;\cdot,\cdot)$ and
${}_{p}F_{q}(\mu;\nu;\cdot,\cdot)$ are entire functions.

  2. (2)

If $p=q+1$, the series ${}_{p}K_{q}(\mu;\nu;\cdot,\cdot)$ and
${}_{p}F_{q}(\mu;\nu;\cdot,\cdot)$ are holomorphic on the domain
$\\{(z,w)\in\mathbb{C}^{n}\times\mathbb{C}^{n}:\left\lVert
z\right\rVert_{\infty}\left\lVert w\right\rVert_{\infty}<1\\}$.

Moreover, the hypergeometric series are holomorphic in the parameters
$(\mu,\nu)$ on the domain

$\set{(\mu,\nu)\in\mathbb{C}^{p}\times\mathbb{C}^{q}\mid\nu_{i}\notin\set{0,k,\ldots,k(n-1)}-\mathbb{N}_{0}\textrm{
for all }i=1,\ldots,n}.$

###### Proof.

It suffices to verify the statements for $\,{}_{p}K_{q}$. From Lemma 3.1(4) we
have $\,\left|L_{\eta}(z)\right|\leq\,L_{\eta}(|z|)\leq
L_{\eta}(\underline{1})\left\lVert
z\right\rVert_{\infty}^{\left|\eta\right|}\,$ and therefore

$\displaystyle S(\mu,\nu;z,w)$
$\displaystyle:=\sum_{\eta\in\mathbb{N}_{0}^{n}}\left|\frac{[\mu]_{\eta_{+}}}{[\nu]_{\eta_{+}}}\right|\cdot\left|\frac{L_{\eta}(z)L_{\eta}(w)}{|\eta|!\,L_{\eta}(\underline{1})}\right|\,\leq\,\sum_{\eta\in\mathbb{N}_{0}^{n}}\left|\frac{[\mu]_{\eta_{+}}}{[\nu]_{\eta_{+}}}\right|\cdot\frac{\|z\|_{\infty}^{|\eta|}\|w\|_{\infty}^{|\eta|}}{|\eta|!}\,L_{\eta}(\underline{1})$
$\displaystyle=\,\sum_{\lambda\in\Lambda_{n}^{+}}\left|\frac{[\mu]_{\lambda}}{[\nu]_{\lambda}}\right|\cdot\frac{\|z\|_{\infty}^{|\lambda|}\|w\|_{\infty}^{|\lambda|}}{|\lambda|!}\,C_{\lambda}(\underline{1}),$

where for the last identity, Lemma 6.1(ii) was used. From Lemmata 3.2 and 6.1
we know that

$C_{\lambda}(\underline{1})\,=\,c_{\lambda}P_{\lambda}(\underline{1})\,\leq\,\frac{|\lambda|!}{\lambda!}Q(\lambda)$
(6.3)

with some polynomial $Q\in\mathcal{P}.$ Therefore, we can find to each
$\epsilon>1$ a constant $C_{\epsilon}>0$ such that $Q(\lambda)\leq
C_{\epsilon}\,\epsilon^{\left|\lambda\right|}$. This gives

$S(\mu,\nu;z,w)\leq\,C_{\epsilon}\sum_{\lambda\in\Lambda_{n}^{+}}\left|\frac{[\mu]_{\lambda}}{[\nu]_{\lambda}}\right|\cdot\frac{\bigl{(}\epsilon\|z\|_{\infty}\|w\|_{\infty}\bigr{)}^{|\lambda|}}{\lambda!}.$
(6.4)

To prove part (1), consider the case $p\leq q.$ In this case, the quotient

$\frac{[\mu]_{\lambda}}{[\nu]_{\lambda}}\,=\,\frac{\prod_{i=1}^{p}[\mu_{i}]_{\lambda}}{\prod_{i=1}^{q}[\nu_{i}]_{\lambda}}$

is of polynomial growth in $\lambda$. To see this, write

$\frac{[\mu]_{\lambda}}{[\nu]_{\lambda}}=\prod\limits_{i=1}^{p}\prod\limits_{j=1}^{n}\frac{\Gamma(\nu_{i}-k(j-1))}{\Gamma(\mu_{i}-k(j-1))}\frac{\Gamma(\mu_{i}+\lambda_{j}-k(j-1))}{\Gamma(\nu_{i}+\lambda_{j}-k(j-1))}.$

By Stirling’s formula we have, locally uniformly in $\mu$ and $\nu$,

$\frac{\Gamma(\mu_{i}+\lambda_{j}-k(j-1))}{\Gamma(\nu_{i}+\lambda_{j}-k(j-1))}\sim(\nu_{i}+\lambda_{j}-k(j-1))^{\nu_{i}-\mu_{i}}\textrm{
for }\lambda_{j}\to\infty.$

Moreover, $|[\nu_{i}]_{\lambda}|\geq 1$ for large $\lambda$. Thus, for each
$\epsilon>1$ there are constant $D>0$ and a compact neighborhood
$K\subseteq\mathbb{C}^{p}\times\mathbb{C}^{q}$ of $(\mu,\nu)$, such that

$\left|\frac{[\mu]_{\lambda}}{[\nu]_{\lambda}}\right|\leq
D\epsilon^{\left|\lambda\right|}\textrm{ for all }(\mu,\nu)\in K.$

Hence, for each $\epsilon>1$, we find a constant $C_{\epsilon}>0$ such that

$\displaystyle S(\mu,\nu;z,w)\,$
$\displaystyle\leq\,C_{\epsilon}\sum_{\lambda\in\Lambda_{n}^{+}}\frac{\bigl{(}\epsilon\|z\|_{\infty}\|w\|_{\infty}\bigr{)}^{|\lambda|}}{\lambda!}\,\leq\,C_{\epsilon}\sum_{\lambda\in\mathbb{N}_{0}^{n}}\frac{\bigl{(}\epsilon\|z\|_{\infty}\|w\|_{\infty}\bigr{)}^{|\lambda|}}{\lambda!}$
$\displaystyle\leq\,C_{\epsilon}\,e^{n\epsilon\|z\|_{\infty}\|w\|_{\infty}}.$

Therefore the $\,{}_{p}K_{q}$-series is converges locally uniformly on
$\mathbb{C}^{n}\times\mathbb{C}^{n}$ and also locally uniformly on the stated
domain of parameters $\mu$ and $\nu$, which proves part (1).

For part (2), observe that for $p=q+1,$ we have

$\frac{[\mu]_{\lambda}}{[\nu]_{\lambda}}\,=\,\frac{\prod_{i=1}^{q}[\mu_{i}]_{\lambda}}{\prod_{i=1}^{q}[\nu_{i}]_{\lambda}}\cdot[\mu_{p}]_{\lambda}\,.$

As in part (1), the first factor is of polynomial growth. Moreover,

$\frac{[\mu_{p}]_{\lambda}}{\lambda!}\,=\,\prod_{j=1}^{n}\frac{\Gamma(\mu_{p}-k(j-1)+\lambda_{j})}{\Gamma(\lambda_{j}+1)\,\Gamma(\mu_{p}-k(j-1))},$

which is of polynomial growth as well. Starting from estimate (6.4), we
therefore obtain that for each $\epsilon>1,$ there is a constant
$C_{\epsilon}>0$ with

$S(\mu,\nu;z,w)\,\leq\,C_{\epsilon}\sum_{\lambda\in\Lambda_{n}^{+}}\bigl{(}\epsilon\|z\|_{\infty}\|w\|_{\infty}\bigr{)}^{|\lambda|}\,\leq\,C_{\epsilon}\,\frac{1}{(1-\epsilon\|z\|_{\infty}\|w\|_{\infty})^{n}}\,.$

This yields the claim. ∎

Note that part (2) of this theorem improves, in the case $w=1$, the results of
[Kan93].

######  Remark 6.6.

For $p=q=0$, one gets the Dunkl kernel and Bessel function of type $A_{n-1},$
respectively. Indeed, Lemma 6.2 just says that

$E(z,w)={}_{0}K_{0}(z,w),$

and symmetrization yields

$J(z,w)={}_{0}F_{0}(z,w),$

which was already noted in [BF98].

######  Remark 6.7.

The proof of Theorem 6.5 shows that for $p\leq q$ and arbitrary $\epsilon>1$
there is a constant $C_{\epsilon}>0$ such that

$\big{|}{}_{p}K_{q}(\mu;\nu;z,w)\big{|}\leq\,S(\mu,\nu;z,w)\,\leq\,C_{\epsilon}\,e^{n\epsilon\|z\|_{\infty}\|w\|_{\infty}}.$
(6.5)

Taking a closer look at the above proof, we see that for $p<q$ this estimate
can be improved. Indeed, consider the quotient

$\frac{[\mu]_{\lambda}}{[\nu]_{\lambda}}=\prod\limits_{j=1}^{p}\frac{[\mu_{j}]_{\lambda}}{[\nu_{j}]_{\lambda}}\cdot\prod\limits_{j={p+1}}^{q}\frac{1}{[\nu_{j}]_{\lambda}}.$

By Stirling’s formula, the first factor is of polynomial growth, and thus of
order $\mathcal{O}({\epsilon_{1}}^{\left|\lambda\right|})$ for arbitrary
$\epsilon_{1}>1,$ while the second factor is of order
$\mathcal{O}(\epsilon_{2}^{-\left|\lambda\right|})$ for arbitrary
$\epsilon_{2}>1$. Under the assumption $p<q$ we therefore obtain the estimate

$\big{|}{}_{p}K_{q}(\mu;\nu;z,w)\big{|}\leq\,S(\mu,\nu;z,w)\,\leq\,C_{\epsilon}\,e^{\epsilon\|z\|_{\infty}\|w\|_{\infty}}.$
(6.6)

for arbitrary $\epsilon>0,$ with some constant $C_{\epsilon}>0$.

The domain of convergence of the hypergeometric series ${}_{p}K_{q}$ and
${}_{p}F_{q}$ and their growth estimates (6.5), (6.6) are important to obtain
from the Laplace transform identities for Jack polynomials in Theorem 3.5
similar Laplace transform identities for the hypergeometric series.

######  Theorem 6.8.

Let $\mu\in\mathbb{C}^{p}$, $\nu\in\mathbb{C}^{q}$ with
$\nu_{i}\notin\set{0,k,\ldots,k(n-1)}-\mathbb{N}_{0}$ for all $i=1,\ldots,n$
and let $\mu^{\prime}\in\mathbb{C}$ with $\text{Re}\,\mu^{\prime}>\mu_{0}$.

  1. (1)

If $p<q$, then for all $z,w\in\mathbb{C}^{n}$ with $\text{Re}\,z>0,$

$\displaystyle\int_{\mathbb{R}_{+}^{n}}E(-z,x)\,{}_{p}K_{q}(\mu;\nu;w,x)$
$\displaystyle\Delta(x)^{\mu^{\prime}-\mu_{0}-1}\omega(x)dx$
$\displaystyle=\Gamma_{n}(\mu^{\prime})\Delta(z)^{-\mu^{\prime}}\,{}_{p+1}K_{q}((\mu^{\prime},\mu);\nu;w,\tfrac{1}{z}).$

  2. (2)

If $p=q$, then part (1) is valid under the condition $\left\lVert
w\right\rVert_{\infty}\cdot\left\lVert\tfrac{1}{\text{Re}\,z}\right\rVert_{\infty}<\tfrac{1}{n}$.

Moreover, both parts remain valid if ${}_{p}K_{q}$ is replaced by
${}_{p}F_{q}$.

###### Proof.

(1) By expanding ${}_{p}K_{q}$ into its defining series, this is immediate
from the Laplace transform identity of Theorem 3.5 by interchanging the order
of summation and integration. We have to justify this interchange. Choose
$\epsilon>0$ such that $\left\lVert w\right\rVert_{\infty}\cdot\left\lVert
1/\text{Re}\,z\right\rVert_{\infty}<\tfrac{1}{\epsilon}$. Under these
conditions the estimates (2.1), (6.5) and (6.6) show that

$\left|E(-x,z)\right|S(\mu,\nu;w,x)\leq
C_{\epsilon}\,e^{-d_{\epsilon}\|x\|_{\infty}}$

with $d_{\epsilon}=\min\limits_{i=1\ldots
n}\text{Re}\,z_{i}-\epsilon\|w\|_{\infty}\,>0.$ Hence, we can apply the
dominated convergence theorem to justify the interchange of summation and
integration, so that part (1) is proven since $\epsilon>0$ was chosen
arbitrarily. Part (2) is obtained in the same way, by choosing $\epsilon>1$
such that $\left\lVert w\right\rVert_{\infty}\cdot\left\lVert
1/\text{Re}\,z\right\rVert_{\infty}<\tfrac{1}{n\epsilon}$. By symmetrization
we get the same identities for ${}_{p}F_{q}$. ∎

We continue with an integral representation which was already observed in
[Mac13, p.39] for the symmetric case, i.e. for ${}_{1}F_{0}$, but only at a
formal level and without any statement on convergence.

######  Corollary 6.9.

Let $\mu\in\mathbb{C}$ with $\text{Re}\,\mu\,>\mu_{0}$. Then the
hypergeometric series ${}_{1}K_{0}(\mu;-z,w)$ has an analytic continuation to
$\,D:=\set{\textrm{Re}\,z>0}\times\set{\textrm{Re}\,w>0}$ which is given by

${}_{1}K_{0}(\mu;-z,w)=\frac{\Delta(z)^{-\mu}}{\Gamma_{n}(\mu)}\int_{\mathbb{R}_{+}^{n}}E(-\tfrac{1}{z},x)E(-w,x)\Delta(x)^{\mu-\mu_{0}-1}\omega(x)dx.$

By symmetrization, the same formula is valid if one replaces ${}_{1}K_{0}$ by
${}_{1}F_{0}$ and the Dunkl kernel by the Bessel function.

###### Proof.

Recall that $\,E(z,w)={}_{0}K_{0}(z,w).$ Then, by Theorem 6.8, the stated
integral formula holds on a suitable open subset of $D.$ Moreover, estimate
(2.1) for the Dunkl kernel shows that the integral exists and defines a
holomorphic function on $D$ by standard theorems on holomorphic parameter
integrals. Hence, analytic continuation finishes the proof. ∎

The following proposition is a generalization of the Euler integral for
hypergeometric functions on symmetric cones ( [FK94, Proposition XV.1.4]) and
can be found as a formal statement in [Mac13, formula (6.21)]. It will be
obtained from Kadell’s [Kad97] generalization of the Selberg integral,

$\int_{[0,1]^{n}}\frac{C_{\lambda}(x)}{C_{\lambda}(\underline{1})}\Delta(x)^{\mu-\mu_{0}-1}\Delta(\underline{1}-x)^{\nu-\mu_{0}-1}\omega(x)dx=\frac{\Gamma_{n}(\mu)\,\Gamma_{n}(\nu)}{\Gamma_{n}(\mu+\nu)}\frac{[\mu]_{\lambda}}{[\mu+\nu]_{\lambda}}$
(6.7)

for all $\lambda\in\Lambda_{+}^{n}$ and $\mu,\nu\in\mathbb{C}$ with
$\text{Re}\,\mu,\text{Re}\,\nu>\mu_{0}$.

######  Proposition 6.10.

Consider $p\leq q+1$ and $\mu^{\prime},\nu^{\prime}\in\mathbb{C}$ with
$Re\,\mu^{\prime},Re(\nu^{\prime}-\mu^{\prime})>\mu_{0}$. Moreover, let
$\mu\in\mathbb{C}^{p}$ and $\nu\in\mathbb{C}^{q}$ with
$\nu_{i}\notin\set{0,k,\ldots,k(n-1)}-\mathbb{N}_{0}$ for all $i=1,\ldots,n$.
Then for arbitrary $w\in\mathbb{C}^{n}$ with the additional condition
$\left\lVert w\right\rVert_{\infty}<1$ in the case $p=q+1,$ one has

$\displaystyle\int_{[0,1]^{n}}{}_{p}F_{q}(\mu;\nu;w,x)\Delta(x)^{\mu^{\prime}-\mu_{0}-1}$
$\displaystyle\Delta(\underline{1}-x)^{\nu^{\prime}-\mu^{\prime}-\mu_{0}-1}\omega(x)dx$
$\displaystyle=\frac{\Gamma_{n}(\mu)\,\Gamma_{n}(\nu-\mu)}{\Gamma_{n}(\nu)}{}_{p+1}F_{q+1}((\mu^{\prime},\mu);(\nu^{\prime},\nu);w,\underline{1}).$

###### Proof.

This is immediate from Kadell’s integral (6.7) after expanding ${}_{p}F_{q}$
into its defining series and changing the order of integration and summation.
The latter is justified since the series ${}_{p}F_{q}(\mu;\nu;w;\cdot)$ is
absolutely bounded on $[0,1]^{n}$ by the estimates in the proof of Theorem 6.5
for $w\in\mathbb{C}^{n}$ and $\left\lVert w\right\rVert_{\infty}<1$ if
$p=q+1$. ∎

The following theorem generalizes Proposition XV.1.2. of [FK94] for
hypergeometric series on symmetric cones.

######  Theorem 6.11.

The Jack polynomials and the hypergeometric series have the following
properties under the action of the Dunkl operator $\Delta(T)$ associated to
the polynomial $\Delta$.

  1. (1)

$\Delta(T)E_{\eta}=c_{\eta}E_{\eta-\underline{1}}\,$ with some constant
$c_{\eta}\in\mathbb{R}.$ Moreover, $c_{\eta}=0$ if $\eta_{i}=0$ for some
$i\in\\{1,\ldots,n\\}.$

  2. (2)

$\eta\mapsto c_{\eta}$ is $\mathcal{S}_{n}$-invariant.

  3. (3)

$\Delta(T)L_{\eta}=d_{\eta}L_{\eta-\underline{1}}$ and
$\Delta(T)C_{\lambda}=d_{\lambda}C_{\lambda-\underline{1}}$, where

$d_{\eta}=\begin{cases}\displaystyle\frac{\left|\eta\right|!}{\left|\eta-\underline{1}\right|!}\,&\textrm{
if }\eta_{i}\neq 0\textrm{ for all }i=1,\ldots,n\\\ \,0\,&\textrm{
otherwise.}\end{cases}.$

  4. (4)

If $p\leq q+1,$ then

$\Delta(T)\,{}_{p}K_{q}(\mu;\nu;w,\cdot)=\frac{[\mu]_{\underline{1}}}{[\nu]_{\underline{1}}}\Delta(w)\,{}_{p}K_{q}(\mu+\underline{1},\nu+\underline{1};w,\cdot)$

for all $w\in\mathbb{C}^{n},$ with the understanding that
$[\mu]_{\underline{1}}=1$ if $p=0$ and $[\nu]_{\underline{1}}=1$ if $q=0$. The
same is valid for ${}_{p}K_{q}$ instead of ${}_{p}F_{q}$.

###### Proof.

(1) From the properties of the Dunkl pairing together with Lemma 3.1 (1) we
can conclude that for compositions $\eta,\kappa\in\mathbb{N}_{0}^{n}$,

$\;[\Delta(T)E_{\eta},E_{\kappa}]=[E_{\eta},\Delta
E_{\kappa}]=[E_{\eta},E_{\kappa+\underline{1}}]=\begin{cases}\,0\,&\textrm{ if
}\eta\neq\kappa+\underline{1}\,;\\\ [E_{\eta},E_{\eta}]>0\,&\textrm{ if
}\eta=\kappa+\underline{1}\,.\end{cases}$

Hence, $\Delta(T)E_{\eta}$ must be a scalar multiple of
$E_{\eta-\underline{1}}$ if $\eta_{i}\geq 1$ for all $i=1,\ldots,n$ and
vanishes otherwise.

(2) Denote again by $\overline{\eta}_{i}$ the eigenvalue of $E_{\eta}$ under
the Cherednik operator $\mathcal{D}_{i}\,.$ It suffices to show that
$c_{\eta}=c_{s_{i}\eta}$ if $\eta_{i}<\eta_{i+1}.$ Due to [For10, Proposition
12.2.1], we then have

$E_{s_{i}\eta}=d_{i}^{\eta}E_{\eta}+s_{i}E_{\eta}$ (6.8)

with the constant

$d_{i}^{\eta}=\frac{k}{\overline{\eta}_{i+1}-\overline{\eta}_{i}}\,.$

It is immediate that
$\overline{\eta+\underline{1}}=\overline{\eta}+\underline{1}$ and therefore
$d_{i}^{\eta+\underline{1}}=d_{i}^{\eta}\,.$ Applying $\Delta(T)$ to equation
(6.8), using $d_{i}^{\eta+\underline{1}}=d_{i}^{\eta}$ and the
$\mathcal{S}_{n}$-equivariance of the Dunkl operators, we obtain from part (1)
that $c_{s_{i}\eta}=c_{\eta}\,.$

(3) As $L_{\eta}$ is a renormalization of $E_{\eta}$, there is a constant
$d_{\eta}$ such that $\Delta(T)L_{\eta}=d_{\eta}L_{\eta-\underline{1}},$ and
$d_{\eta}=0$ if $\eta_{i}=0$ for some $i.$ Since
$\,\eta\mapsto\frac{\left|\eta\right|!}{\left|\eta-\underline{1}\right|!}$ is
$\mathcal{S}_{n}$-invariant and
$\,C_{\lambda}=\sum_{\eta\in\mathcal{S}_{n}\lambda}L_{\eta}\,,$ it suffices to
verify the stated value of $d_{\eta}$ with $|\eta|\geq n.$ Recall that $T_{i}$
acts as $\partial_{i}\,$ on symmetric polynomials. We therefore conclude that
for $m\in\mathbb{N}_{0}$ with $m\geq n,$

$\displaystyle\quad m\cdots(m-n+1)$
$\displaystyle\\!\\!\sum\limits_{\begin{subarray}{c}\eta\in\mathbb{N}_{0}^{n}:\\\
\left|\eta\right|=m-n\end{subarray}}L_{\eta}(x)\,=\,m\cdots(m-n+1)(x_{1}+\ldots+x_{n})^{m-n}$
$\displaystyle=\Delta(T)(x_{1}+\ldots+x_{n})^{m}=\sum\limits_{\begin{subarray}{c}\eta\in\mathbb{N}_{0}^{n}:\\\
\left|\eta\right|=m\end{subarray}}\Delta(T)L_{\eta}\,=\,\sum\limits_{\begin{subarray}{c}\eta\in\mathbb{N}_{0}^{n}:\\\
\left|\eta\right|=m\end{subarray}}d_{\eta}L_{\eta-\underline{1}}.$

Equating the coefficients proves the stated formula for $d_{\eta}$.

(4) This is an immediate consequence of part (3) by expanding the
hypergeometric series. One has to perform an index shift
$\eta\mapsto\eta+\underline{1}$ after applying $\Delta(T)$ and the identity of
part (3) together with

$\frac{L_{\eta}(w)}{L_{\eta}(\underline{1})}=\Delta(w)\frac{L_{\eta-\underline{1}}(w)}{L_{\eta-\underline{1}}(w)}$

and
$[\theta]_{\eta_{+}}=[\theta+1]_{\eta_{+}-\underline{1}}[\theta]_{\underline{1}}$.
∎

##  7\. A Post-Widder inversion formula for the Dunkl-Laplace transform

We consider again the Dunkl setting of type $A_{n-1}$ with multiplicity $k\geq
0$ and keep our previous notations. As shown in [Rös20], the Dunkl-Laplace
transform

$\mathcal{L}f(z)=\int_{\mathbb{R}_{+}^{n}}f(x)E(-z,x)\omega(x)dx$

satisfies the following Cauchy inversion theorem: Let $f\in
L_{loc}^{1}(\mathbb{R}_{+}^{n})$ such that $\mathcal{L}f(\underline{s})$
exists for some $s\in\mathbb{R}$ (then $\mathcal{L}f(z)$ also exists for all
$z\in\mathbb{C}^{n}$ with $\text{Re}\,z=\underline{s}\,$). Assume further that
$y\mapsto\mathcal{L}f(\underline{s}+iy)\in L^{1}(\mathbb{R}^{n},\omega).$ Then
$f$ has a continuous representative $f_{0}$, and

$\frac{(-i)^{n}}{c_{k}^{2}}\int_{\text{Re}\,z=\underline{s}}\mathcal{L}f(z)E(x,z)\omega(z)dz=\,\begin{cases}f_{0}(x)&\text{
for }x\in\mathbb{R}_{+}^{n};\\\ \,0&\text{ otherwise},\end{cases}$

with the constant $\,c_{k}=\int_{\mathbb{R}^{n}}e^{-|x|^{2}/2}\omega(x)dx.$
For the classical Laplace transform

$Lf(z)=\int_{0}^{\infty}f(x)e^{-zx}dx,\quad f\in L_{loc}^{1}(\mathbb{R}_{+}),$

a further well-known inversion theorem is the Post-Widder inversion formula
(see e.g. [ABHN01]): Assume that $f\in L_{loc}^{1}(\mathbb{R}_{+})$ has a
finite abscissa of convergence and is continuous in $\xi\in\mathbb{R}_{+}$.
Then

$f(\xi)=\lim_{\nu\to\infty}\frac{(-1)^{\nu}}{\nu!}\Bigl{(}\frac{\nu}{\xi}\Bigr{)}^{\nu+1}(Lf)^{(\nu)}\Bigl{(}\frac{\nu}{\xi}\Bigr{)}.$

In this section, we prove a Post-Widder inversion formula for the Dunkl-
Laplace transform, which is the counterpart to a result of Faraut and Gindikin
[FG90] in the setting of symmetric cones.

######  Theorem 7.1 (Post-Widder inversion formula for $\mathcal{L}$).

Let $f:\mathbb{R}_{+}^{n}\to\mathbb{C}$ be measurable and bounded, and suppose
that $f$ is continuous at $\xi\in\mathbb{R}_{+}^{n}\,.$ Then

$f(\xi)=\lim_{\nu\to\infty}\frac{(-1)^{n\nu}}{\Gamma_{n}(\nu+\mu_{0}+1)}\Delta\Bigl{(}\frac{\nu}{\xi}\Bigr{)}^{\nu+\mu_{0}+1}\bigl{(}\Delta(T)^{\nu}(\mathcal{L}f)\bigr{)}\Bigl{(}\frac{\nu}{\xi}\Bigr{)}.$

The idea of proof for this theorem is similar to [FG90]. It was elaborated to
some extent by Frederik Hoppe in his master thesis [Hop20], which was
supervised by the second author of this paper. A fundamental ingredient is
Levy’s continuity theorem for the Dunkl transform. Let us recall this for the
reader’s convenience. Denote by $M_{b}^{+}(\mathbb{R}^{n})$ the space of
positive bounded Borel measures on $\mathbb{R}^{n}$. The Dunkl transform of
$\mu\in M_{b}^{+}(\mathbb{R}^{n})$ (associated with $A_{n-1}$ and multiplicity
$k$) is given by

$\widehat{\mu}(\xi)=\widehat{\mu}^{\,k}(\xi)=\,\int_{\mathbb{R}^{n}}E(-i\xi,x)d\mu(x),\quad\xi\in\mathbb{R}^{n}.$

Note that $\widehat{\mu}\in C_{b}(\mathbb{R}^{n}),$ since $|E(-i\xi,x)|\leq 1$
for all $\xi,x\in\mathbb{R}^{n}$. The Dunkl transform is injective on
$M_{b}^{+}(\mathbb{R}^{n}),$ see [RV98]. The following is the essential part
of Levy’s continuity theorem for the Dunkl transform.

######  Lemma 7.2 ([RV98]).

Let $(\mu_{\nu})_{\nu\in\mathbb{N}}\subseteq M_{b}^{+}(\mathbb{R}^{n})$ such
that the sequence $(\widehat{\mu}_{\nu})_{\nu\in\mathbb{N}}$ converges
pointwise to a function $\varphi:\mathbb{R}^{n}\to\mathbb{C}$ which is
continuous at $0$. Then there exists a unique $\mu\in
M_{b}^{+}(\mathbb{R}^{n})$ with $\,\widehat{\mu}_{\nu}=\varphi$, and
$(\mu_{\nu})_{\nu\in\mathbb{N}}$ converges to $\mu$ weakly.

###### Proof of Theorem 7.1.

We consider on $\mathbb{R}_{+}^{n}$ the functions

$h_{\nu}(x):=E\bigl{(}-\tfrac{\nu}{\xi},x\bigr{)}\,\Delta(x)^{\nu},\quad\nu\in\mathbb{N}.$

By estimate (2.1), the Laplace transform

$\mathcal{L}h_{\nu}(z)=\int_{\mathbb{R}_{+}^{n}}E(-z,x)E\bigl{(}-\tfrac{\nu}{\xi},x\bigr{)}\Delta(x)^{\nu}\omega(x)dx$

exists for all $z\in\mathbb{C}^{n}$ with $\text{Re}\,z\geq 0.$ For such $z$,
put $\nu(z):=\max_{i}\lceil\|z\|_{\infty}\,\xi_{i}\rceil\in\mathbb{N}.$ Then
for $\nu>\nu(z),$ we calculate

$\displaystyle\mathcal{L}h_{\nu}(z)\,$
$\displaystyle=\,\int_{\mathbb{R}_{+}^{n}}\Bigl{(}\sum_{\eta\in\mathbb{N}_{0}^{n}}\frac{L_{\eta}(-z)L_{\eta}(x)}{|\eta|!\,L_{\eta}(\underline{1})}\Bigr{)}E\bigl{(}-\tfrac{\nu}{\xi},x\bigr{)}\Delta(x)^{\nu}\omega(x)dx$
$\displaystyle=\,\sum_{\eta\in\mathbb{N}_{0}^{n}}\frac{L_{\eta}(-z)}{|\eta|!\,L_{\eta}(\underline{1})}\int_{\mathbb{R}_{+}^{n}}L_{\eta}(x)E\bigl{(}-\tfrac{\nu}{\xi},x\bigr{)}\Delta(x)^{\nu}\omega(x)dx$
$\displaystyle=\,\sum_{\eta\in\mathbb{N}_{0}^{n}}\frac{L_{\eta}(-z)}{|\eta|!\,L_{\eta}(\underline{1})}\,\Gamma_{n}(\eta_{+}+\nu+\mu_{0}+1)\,L_{\eta}\bigl{(}\tfrac{\xi}{\nu}\bigr{)}\,\Delta\bigl{(}\tfrac{\nu}{\xi}\bigr{)}^{-\nu-\mu_{0}-1}.$

Here the interchange of the sum and the integral is justified by the dominated
convergence theorem, because $\,|L_{\eta}(-z)|\leq
L_{\eta}(\|z\|_{\infty}\cdot\underline{1})$ and therefore

$\displaystyle
E\bigl{(}-\tfrac{\nu}{\xi},x\bigr{)}\sum_{\eta\in\mathbb{N}_{0}^{n}}\frac{|L_{\eta}(-z)L_{\eta}(x)|}{|\eta|!\,L_{\eta}(\underline{1})}\,$
$\displaystyle\leq
E\bigl{(}-\tfrac{\nu}{\xi},x\bigr{)}E(\|z\|_{\infty}\cdot\underline{1},x)$
$\displaystyle=\,E\bigl{(}-\tfrac{\nu}{\xi}+\|z\|_{\infty}\cdot\underline{1},x\bigr{)}.$

This decays exponentially on $\mathbb{R}_{+}^{n}$, since
$-\nu/\xi+\|z\|_{\infty}\cdot\underline{1}<0$ by our assumption on $\nu.$ Thus
for $\nu\geq\nu(z),$

$f_{\nu}(z):=\frac{\Delta\bigl{(}\frac{\nu}{\xi}\bigr{)}^{\nu+\mu_{0}+1}}{\Gamma_{n}(\nu+\mu_{0}+1)}\,\mathcal{L}h_{\nu}(z)\,=\,\sum_{\eta\in\mathbb{N}_{0}^{n}}c_{\nu}(\eta)\cdot\frac{L_{\eta}(-z)L_{\eta}(\xi)}{|\eta|!\,L_{\eta}(\underline{1})}$
(7.1)

with the coefficients

$c_{\nu}(\eta)=\frac{[\nu+\mu_{0}+1]_{\eta_{+}}}{\nu^{|\eta|}}\,=\,\prod_{j=1}^{n}\Bigl{(}1+\frac{1+k(n-j)}{\nu}\Bigr{)}_{\lambda_{j}},\quad\lambda=\eta_{+}\,.$

They satisfy

$\lim_{\nu\to\infty}c_{\nu}(\eta)=1\quad\text{ for fixed }\,\eta,$

and it follows that

$\lim_{\nu\to\infty}f_{\nu}(z)\,=\,\sum_{\eta\in\mathbb{N}_{0}^{n}}\frac{L_{\eta}(-z)L_{\eta}(\xi)}{|\eta|!\,L_{\eta}(\underline{1})}\,=\,E(-z,\xi).$
(7.2)

We still have to justify that the limit $\nu\to\infty$ may be taken inside the
sum in (7.1). For this, note that $\nu\mapsto c_{\nu}(\eta)$ is monotonically
decreasing. Hence for $\nu\geq\nu(z),$ the series on the right-hand side of
(7.1) is dominated by the convergent series

$\sum_{\eta\in\mathbb{N}_{0}^{n}}c_{\nu(z)}(\eta)\,\frac{L_{\eta}(\|z\|_{\infty}\cdot\underline{1})\,L_{\eta}(\xi)}{|\eta|!\,L_{\eta}(\underline{1})}\,=\,f_{\nu(z)}(-\|z\|_{\infty}\cdot\underline{1})<\infty,$

which justifies the above limit. We now consider the measures

$dm_{\nu}(x):=\frac{\Delta\bigl{(}\frac{\nu}{\xi}\bigr{)}^{\nu+\mu_{0}+1}}{\Gamma_{n}(\nu+\mu_{0}+1)}\cdot
1_{\mathbb{R}_{+}^{n}}(x)E\bigl{(}-\tfrac{\nu}{\xi},x\bigr{)}\Delta(x)^{\nu}\omega(x)dx\,\in
M_{b}^{+}(\mathbb{R}^{n}).$

Due to Theorem 3.5, $m_{\nu}$ is actually a probability measure on
$\mathbb{R}^{n}.$ Formula (7.2), considered for arguments $z\in
i\mathbb{R}^{n}$, shows that the Dunkl transforms satisfy

$\widehat{m_{\nu}}\to\widehat{\delta_{\xi}}\quad\text{ pointwise on
}\mathbb{R}^{n},$

where $\delta_{\xi}$ denotes the point measure in $\xi$. Levy’s continuity
theorem (Lemma 7.2) now implies that $\,m_{\nu}\to\delta_{\xi}$ weakly. Thanks
to the Portemanteau theorem ([Kle14]) we even get

$\lim_{\nu\to\infty}\int_{\mathbb{R}^{n}}g\,dm_{\nu}=\int_{\mathbb{R}^{n}}g\,d\delta_{\xi}\,\,=\,g\bigl{(}\xi\bigr{)}$

for all measurable bounded functions $g:\mathbb{R}^{n}\to\mathbb{C}$ which are
continuous at $\xi$. Now suppose $f:\mathbb{R}_{+}^{n}\to\mathbb{C}$ is
measurable, bounded and continuous at $\xi.$ Extend $f$ by zero to
$\mathbb{R}^{n}.$ Then

$\,\frac{\Delta\bigl{(}\frac{\nu}{\xi}\bigr{)}^{\nu+\mu_{0}+1}}{\Gamma_{n}(\nu+\mu_{0}+1)}\int_{\mathbb{R}_{+}^{n}}f(x)E\bigl{(}-\tfrac{\nu}{\xi},x\bigr{)}\Delta(x)^{\nu}\omega(x)dx\,=\,\int_{\mathbb{R}^{n}}fdm_{\nu}\,\to
f(\xi).$

But in view of to Lemma 2.1 the integral on the left-hand side can be written
as

$\mathcal{L}(\Delta^{\nu}f)\bigl{(}\tfrac{\nu}{\xi}\bigr{)}=\,\bigl{(}\Delta(-T)\bigr{)}^{\nu}(\mathcal{L}f)\bigl{(}\tfrac{\nu}{\xi}\bigr{)},$

which finishes the proof. ∎

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