2Dunkloperators 1Introduction Dunkl-TypeOperatorswithProjectionsTermsAssociatedtoOrthogonalSubsystemsinRootsSystem

Dunkl-Type Operators with Projections Terms Associated to Orthogonal Subsystems

in Roots System

Fethi BOUZEFFOUR

Department of Mathematics, King Saudi University, College of Sciences, P.O. Box 2455 Riyadh 11451, Saudi Arabia

E-mail: [email protected]

Received April 24, 2013, in final form October 16, 2013; Published online October 23, 2013 http://dx.doi.org/10.3842/SIGMA.2013.064

Abstract. In this paper, we introduce a new differential-difference operatorT_ξ (ξ∈R^N) by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator betweenT_ξ and the directional derivative∂_ξ. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this operator.

Key words: special functions; differential-difference operators; integral transforms 2010 Mathematics Subject Classification: 33C15; 33D52; 35A22

1 Introduction

In a series of papers [3,4,5,6], C.F. Dunkl builds up the framework for a theory of differential- difference operators and special functions related to root systems. Beside them, there are now various further Dunkl-type operators, in particular the trigonometric Dunkl operators of Heckman [7, 8], Opdam [14], Cherednik [2], and the important q-analogues of Macdonald and Cherednik [13], see also [1,11].

The main objective of this paper is to present a new class of differential-difference opera- tors T_ξ, ξ ∈ R^N with the help of orthogonal projections related to orthogonal subsystems in root systems. In other words, our operators follow from Dunkl operator after replacing the usual reflections that exist in the definition of the operator with their corresponding ortho- gonal projections. Several problems related to the Dunkl theory arise in the setting of our operators, in particular, commutativity of {T_ξ, ξ ∈ R^N} and the existence of the intertwining operators.

The outline of the content of this paper is as follows. In Section2, we collect some definitions and results related to root systems and Dunkl operators which will be relevant for the sequel.

In Section 3, we introduce new Differential-difference operatorsT_ξ and we prove the first main result. In Section 4, we give an explicit formula for the intertwining operator between Tξ and the directional derivative. In Section5, we study the one variable case. Finally, in Section6we study the cases of orthogonal subsets in root systems of type A_N−1 and B_N.

2 Dunkl operators

Let us begin to recall some results concerning the root systems and Dunkl operators. A useful reference for this topic is the book by Humphreys [9]. Let α ∈ R^N\{0}, we denote by sα the

reflection onto the hyperplane orthogonal to α; that is, s_α(x) =x−2hx, αi

|α|² α,

where h·,·idenotes the Euclidean scalar product on R^N, and |x|=p hx, xi.

A root system is a finite setR of nonzero vectors inR^N such that for any α∈R one has sα(R) =R, and R∩Rα={±α}.

A positive subsystem R₊ is any subset of R satisfying R = R₊∪ {−R₊}. The Weyl group W = W(R) (or real finite reflection group) generated by the root system R ⊂ R^N is the subgroup of orthogonal group O(N) generated by {s_α, α∈R}. A multiplicity function onR is a complex-valued functionκ:R→Cwhich is invariant under the Weyl groupW, i.e.,

κ(α) =κ(gα), ∀α∈R, ∀g∈W.

Letξ ∈R^N, the Dunkl operator D_ξ associated with the Weyl group W(R) and the multiplicity functionκ, is the first order differential-difference operator:

(D_ξf)(x) =∂_ξf(x) + X

α∈R+

κ(α)hα, ξif(x)−f(s_αx)

hx, αi . (1)

Here ∂_ξ is the direction derivative corresponding to ξ and sα is the orthogonal reflection onto the hyperplane orthogonal toα.

The Dunkl operatorD_ξ is a homogeneous differential-difference operator of degree −1. By theW-invariance of the multiplicity function κ, we have

g⁻¹◦ D_ξ◦g=D_gξ, ∀g∈W(R), ξ ∈R^N.

The remarkable property of the Dunkl operators is that the family {D_ξ, ξ ∈ R^N} generates a commutative algebra of linear operators on theC-algebra of polynomial functions.

3 Operators of Dunkl-type

LetR be a root system. A subset R⁰ of R is called a subsystem ofR if it satisfies the following conditions:

i) If α∈R⁰, then−α∈R⁰;

ii) Ifα, β∈R⁰ andα+β ∈R, thenα+β ∈R⁰.

A subsystem R⁰ of a root system R in R^N consisting of pairwise orthogonal roots is called orthogonal subsystem. In this case the related Weyl group W(R⁰) is a subgroup of Z^N₂ . For a vector α∈R^N \ {0}, we write

τ_α(x) =x−hx, αi

|α|² α, x∈R^N,

for the orthogonal projection onto the hyperplane (Rα)^⊥ ={x,hx, αi= 0}, so that the reflec- tion s_α with respect to hyperplane orthogonal toα is related toτ_α by

τα = 1

2(1 +sα).

The hyperplane (Rα)^⊥ is the invariant set of τα. If hα, βi = 0, then the orthogonal projec- tions τ_α and τ_β commute. The conjugate of orthogonal projection onto a hyperplane is again an orthogonal projection onto a hyperplane: suppose u∈O(N) andα∈R^N\{0} then

uτ_αu⁻¹=τ_uα.

Let R be a root system and R⁰ a positive orthogonal subsystem of R. For ξ ∈R^N, we define the differential-difference operator Tξ by

(T_ξf)(x) =∂_ξf(x) + X

α∈R⁰

κ(α)hα, ξif(x)−f(τ_αx)

hx, αi . (2)

where κ is a multiplicity function onR⁰. For j= 1, . . . , N denotesT_ej by T_j. The operator T_ξ can be considered as a deformation of the usual directional derivatives and when κ = 0, the operator T_ξ reduces to the corresponding directional derivative. Furthermore, there is overlap between the notations (2) and (1). In fact, the operator (2) follows from Dunkl operator after replacing the reflections terms that exist in (1) by orthogonal projections terms.

Example 1. In the rank-one case, the root system is of typeA1and the corresponding reflection sand orthogonal projection τ are given by

s(x) =−x, τ(x) = 1

2(1 +s)(x) = 0.

The Dunkl-type operator Tκ associated with the projection τ and the multiplicity parametersκ (κ∈C) is given by

Tκf(x) =f⁰(x) +κf(x)−f(τ(x))

x =f⁰(x) +κf(x)−f(0)

x .

Example 2. Let R ={±(e₁ ±e₂),±e₁,±e₂} be a root system of type B₂ in the 2-plane and R⁰ ={e₁±e₂}be a positive orthogonal subsystem inR. The related Dunkl-type operators toR⁰ and to the positive parameters (κ₁, κ₂) are given by

T1=∂x+κ1

f(x, y)−f((x+y)/2,(x+y)/2)

x−y +κ2

f(x, y)−f((x−y)/2,(x−y)/2)

x+y ,

T₂=∂_y−κ₁f(x, y)−f((x+y)/2,(x+y)/2)

x−y +κ₂f(x, y)−f((x−y)/2,(x−y)/2)

x+y .

We denote by Π^N the space of polynomials and by Π^N_n the subspace of homogenous polyno- mials of degree n.

LetR⁰={α₁, . . . , αn} be a positive orthogonal subsystem of a root system R. Consider the operatorρ_i defined on Π^N by

(ρif)(x) = f(x)−f(ταix)

hx, αi , i= 1, . . . , n.

It follows from the equality (ρ_jf)(x) =− 1

|α_j|² Z 1

0

∂_αjf

x−thx, α_ji

|α_j|² α_j

dt

that Tξ is a homogeneous operator of degree −1 on Π^N, that is, Tξf ∈ Π^N_n−1, forf ∈Π^N_n, and leaves S(R) (S(R) is the Schwartz space of rapidly decreasing functions on R) invariant.

Proposition 1. The operators ρi (i= 1, . . . , n) have the following properties:

i) for i, j= 1, . . . , n, we have [ρi, ρj] = 0;

ii) ifα is an orthogonal vector toα_i, then[∂_α, ρ_i] = 0, where the commutator of two operators A, B is defined by [A, B] :=AB−BA.

The family{α₁, . . . , αn}is orthogonal, then there exist scalarsξ1, . . . , ξnand a vectorξb∈R^N orthogonal to the subspace Rα₁⊕ · · · ⊕Rα_n such that

ξ =

n

X

i=1

ξ_iα_i+ξ.b

This allows us to decompose the operator Tξ (2) associated with R⁰ and the multiplicity para- meters (κ1, . . . , κn) in a unique way in the form

T_ξ=

n

X

i=1

ξ_iT_αi+∂

ξb.

We now have all ingredients to state and prove the first main result of the paper.

Theorem 1. Let ξ, η∈R^N, then [T_ξ, T_η] = 0.

Proof . A straightforward computation yields [T_ξ, T_η] =

n

X

i,j=1

ξ_iη_j[T_αi, T_αj] + [∂

bξ, ∂

ηb] +

n

X

i=1

ξ_i[T_αi, ∂

bη]−η_i[T_αi, ∂

ξb].

On the other hand,

[T_αi, T_αj] = [∂_αi+κ_ikα_ikρ_i, ∂_αj+κ_jkα_jkρ_j]

= [∂_αi, ∂_αj] +κ_jkα_jk[∂_αi, ρ_j]−κ_ikα_ik[∂_αj, ρ_i] +κ_iκ_jkα_ikkα_jk[ρ_i, ρ_j], and

[T_αi, ∂

ξb] = [∂_αi, ∂

ξb] +κ_ikα_ik[ρ_i, ∂_ξ].

From Proposition1, we get

[T_αi, T_αj] = 0 and [T_αi, ∂

ξb] = 0.

This proves the result.

One important consequence of the Theorem 1, is that the operators T_α1, . . . , T_αm generate a commutative algebra.

4 Intertwining operator

In this section, we give an intertwining operator between T_ξ and the directional derivative ∂_ξ. Consider a positive orthogonal subsystem R⁰ = {α₁, . . . , αn} composed of n vectors in a root system R, and κ = (κ₁, . . . , κ_n) ∈ Cⁿ and ξ ∈ R^N. The associated Dunkl-type operator T_ξ with R⁰ and κtakes the form

(T_ξf)(x) =∂_ξf(x) +

n

X

j=1

κ_jhα_j, ξif(x)−f(ταjx) hx, α_ji .

Let h:Rⁿ×R^N →R^N be the function defined by h(t, x) =x+

n

X

j=1

(tj−1)hx, α_ji

|α_j|² αj, where t= (t₁, . . . , t_n)∈Rⁿand x∈R^N.

We define

χ_κ(f)(x) = 1 Γ(κ)

Z

[0,1]ⁿ

f(h(t, x))w(t) dt, (3)

where w(t) = Qn

j=1

(1−t_j)^κj−1 and Γ(κ) = Qn

j=1

Γ(κ_j).

Theorem 2. Let f ∈C^∞(R^N), then we have T_ξ◦χκf(x) =χκ◦∂_ξf(x).

Proof . For j = 1, . . . , n, we denote by θ_j the orthogonal projection in Rⁿ with respect to the hyperplane (Rej)^⊥ orthogonal to the vector ej of the canonical basis (e1, . . . , en) of Rⁿ. The orthogonal projection θj acts onRⁿ as

θ_j(t) = (t₁, . . . , tj−1,0, t_j+1, . . . , t_n).

The system R is orthogonal, then forj = 1, . . . , n, we have h(t, ταjx) =ταjx+

n

X

k=1

(tk−1)hτ_αjx, α_ki

|α_k|² αk

=x−hx, α_ji

|α_j|² αj +

n

X

k=1,k6=j

(tk−1)hx, α_ki

|α_k|² αk =h(θjt, x).

Let f ∈C^∞(R^N) and ξ∈R^N. The mapping x→h(t, x) is linear on R^N, then we can write

∂_ξ(f(h(t, x))) =∂_h(t,ξ)f(h(t, x)) =∂_ξf(h(t, x)) +

n

X

j=1

(t_j−1)hξ, α_ji

|α_j|² ∂_αjf(h(t, x)).

Hence,

∂ξχκ(f)(x) = 1 Γ(κ)

Z

[0,1]ⁿ

∂ξ(f(h(t, x)))w(t) dt= 1 Γ(κ)

Z

[0,1]ⁿ

∂ξf(h(t, x))w(t) dt

+ 1

Γ(κ)

n

X

j=1

hξ, α_ji

|α_j|² Z

[0,1]ⁿ

(tj−1)∂αjf(h(t, x))w(t) dt.

Since we can write

∂tjf(h(t, x)) = hx, α_ki

|α_k|² ∂αjf(h(t, x)) and

Z ₁

0

(1−t_j)^κj∂_tjf(h(t, x)) dt=−f(h(θ_j(t), x)) +κ_j Z ₁

0

(1−t_j)^κj−1f(h(t, x)) dt,

we are lead to Z

[0,1]ⁿ

∂_αjf(h(t, x))(t_j−1)w(t) dt= |α_j|² hx, α_ji

Z

[0,1]ⁿ

∂_tjf(h(t, x))(t_j−1)w(t) dt

=κj

|α_j|² hx, α_ji

Z

[0,1]ⁿ

(f(h(θj(t), x))−f(h(t, x)))w(t) dt

=−κ_jΓ(κ) |α_j|²

hx, α_ji χ_κ(f)(x)−χ_κ(f)(τ_αjx) . This, combined with the last expression of∂_ξ(χ_κf)(x), yields

∂_ξχ_κ(f)(x) =χ_κ(∂_ξf)(x)−

n

X

j=1

κ_jhξ, α_jiχ_κ(f)(x)−χ_κ(f)(τ_jx) hx, α_ji . Therefore,

T_ξ(χ_κf)(x) =χ_κ(∂_ξf)(x).

5 The one variable case

The specialization of this theory to the one variable case has its own interest, because everything can be done there in a much more explicit way and new results for special functions in one variable can be obtained. In this setting there is only one Dunkl-type operator Tκ associated up to scaling and it equals to

T_κf(x) =f⁰(x) +κf(x)−f(0)

x . (4)

This operator leaves the space of polynomials invariant and acts on the monomials as Tκ1 = 0, Tκxⁿ= (n+κ)xⁿ⁻¹, n= 1,2, . . . .

Its square is given by T_κ²f(x) =f⁰⁰(x) +2κ

x f⁰(x) +κ(κ−1)f(x)−f(0)

x² −κ(κ+ 1) x f⁰(0).

Consider the confluent hypergeometric function (see [15,§ 7.1]) M(a, b;z) =

∞

X

n=0

(a)_n (b)n

zⁿ n!,

where (a)_n is the Pochhammer symbol defined by (a)_n= Γ(a+n)

Γ(a) .

This is a solution of the confluent hypergeometric differential equation zy⁰⁰(z) + (b−z)y⁰(z) =ay(z).

This function possesses the following Poisson integral representation (see [15,§ 7.1]) M(a, b;z) = Γ(b)

Γ(a)Γ(b−a) Z 1

0

t^a−1(1−t)^b−a−1e^ztdt, <(b)><(a)>0. (5)

Theorem 3. For λ∈C and κ >−1, the problem

T_κf(x) =iλf(x), f(0) = 1, (6)

has a unique analytic solution Mκ(iλx) given by

M_κ(iλx) =M(1, κ+ 1;iλx). (7)

Proof . Searching a solution of (6) in the form f(z) =

∞

P

n=0

anxⁿ. Replacing in (6), we obtain

∞

X

n=0

(n+ 1 +κ)an+1xⁿ=iλ

∞

X

n=0

anxⁿ.

Thus,

an+1= iλ

n+ 1 +κan and an= (iλ)ⁿ

(κ+ 1)_n.

Remark 1. Multiply the equation (6) by x and differentiating both sides, we see that a func- tion u of class C² on R, is a solution of the equation (6), if and only if, it is a solution of the generalized eigenvalue problem

xu⁰⁰+ (κ+ 1)u⁰ =iλ(xu⁰+u).

Proposition 2. The function Mκ(z) defined by Mκ(z) = Mκ(z)

Γ(κ+ 1) =

∞

X

n=0

zⁿ

Γ(κ+ 1 +n) (8)

satisfies the following properties:

(i) Mκ(z) is analytic in κ and z;

(ii) M0(z) =e^z;

(iii) for <(κ)>0, the function M_κ(z), possesses the integral representation Mκ(z) = 1

Γ(κ) Z 1

0

(1−t)^κ−1e^ztdt;

(iv) for <(κ)>0, we have M⁽ⁿ⁾_κ (z)

≤ |z|ⁿe^<(z), n∈N, z∈C, in particular,

|M_κ(iλx)| ≤1, λ, x∈R; (v) for <(κ)>0, and all x∈R^∗,

λ→+∞lim Mκ(iλx) = 0.

Proof . (i) and (ii) are immediate. (iii) follows from (5). For n∈N, we have M⁽ⁿ⁾_κ (z) = zⁿ

Γ(κ) Z 1

0

(1−t)^κtⁿe^ztdt.

So we find

|M⁽ⁿ⁾_κ (z)| ≤ |z|ⁿ Γ(κ)

Z 1 0

(1−t)^κe^<(z)tdt≤ |z|ⁿe^<(z).

This proves (iv). (v) follows from (iii) and the Riemann–Lebesgue lemma.

Definition 1. We define the Kummer transform on L¹(R) by

∀λ∈R, F_κ(f)(λ) = Z

R

f(x)M_κ(iλx)(x) dx.

When κ= 0, the transformationF₀ reduces to the usual Fourier transform F that is given by F(f)(λ) =

Z

R

f(x)e^iλxdx.

Theorem 4. Let f be a function in L¹(R) then F_κ(f) belongs to C₀(R), where C₀(R) is the space of continuous functions having zero as limit at the infinity. Furthermore,

kF_κ(f)k∞≤ kfk₁.

Proof . It’s clear that F_κ(f) is a continuous function on R. From Proposition 2, we get for all x∈R^∗,

λ→∞lim f(x)M_κ(iλx) = 0 and |f(x)M_κ(iλx)| ≤ |f(x)|.

Sincef is inL¹(R), we conclude by using the dominated convergence theorem thatF_κ(f) belongs toC0(R) and

kF_κ(f)k_∞≤ kfk₁.

We now turn to exhibit a relationship between the Kummer transform and the Fourier transform.

The crucial idea is to use the intertwining operator χ_κ. We denote by C^∞(R) the space of infinitely differentiable functionsf onR, provided with the topology defined by the semi norms

kfk_n,a = sup

0≤k≤n x∈[−a,a]

f^(k)(x)

, a >0, n∈N.

In the rank-one case the intertwining operator (3) becomes (χκf)(x) = 1

Γ(κ) Z 1

0

(1−t)^κ−1f(tx) dt. (9)

This operator is a particular case of the so called Erd´elyi–Kober fractional integral I^γ,δ, which is given by (see [10])

(I^γ,δf)(x) = 1 Γ(δ)

Z 1 0

(1−t)^δ−1t^γf(tx) dt, δ >0, γ ∈R.

It was shown in [12,§ 3], that the Erd´elyi–Kober fractional integral has a left-inverse

D^γ,δI^γ,δf =f, f ∈C^∞(R), (10)

where D^γ,δ =

n

Y

k=1

γ+k+x d dx

I^γ+δ,n−δ,

and n=dδe (dδe denotes the ceiling function the smallest integer≥δ).

As a consequence of Theorem 2, we deduce that the operator χκ (9) has the fundamental intertwining property

Tκ◦χκ =χκ◦ d dx.

We regard it as a second main result since it allows us to move from the complicated operatorT_κ

defined in (4) to the simple derivative operator _dx^d .

Theorem 5. Let κ > 0, the operator χ_κ is a topological isomorphism from C^∞(R) onto itself and its inverse χ⁻¹_κ is given for all f ∈C^∞(R) by

χ⁻¹_κ f(x) =D^0,κf(x) =

n

Y

j=1

j+x d dx

(I^κ+1,n−κf)(x), where n=dκe.

Proof . Let a > 0 and f ∈C^∞(R). For x ∈[0, a], t ∈ [0,1] and l ∈ N, we have the following estimate

|t^l(1−t)^κ−1f^(l)(xt)| ≤ kfk_l,a(1−t)^κ−1 and

Z 1 0

(1−t)^κ−1dt= 1 κ. By the theorem of derivation under the integral sign, we can prove that

χ_κf ∈C^∞(R) and kχ_κ(f)k_l,a ≤ 1

Γ(κ+ 1)kfk_l,a.

Then χ_κ is a linear continuous mapping from C^∞(R) onto its self. From formula (10) the operator

D^0,κ=

n

Y

j=1

j+x d dx

◦I^κ+1,n−κ

is a left-inverse of χ_κ. This shows that χ_κ is injective and D^0,κ is surjective. So it suffices to prove that D^0,κ is injective.

Let f be a function in C^∞(R) such that D^0,κf = 0. Then the function g = I^κ+1,n−κf ∈ C^∞(R) is a solution of the linear differential equation

n

Y

j=1

1 +j+x d dx

y(x) = 0.

Since, the last differential equation has a unique C^∞-solution, which is equal to y(x) = 0, it follows thatg= 0.

From (10) the operatorI^κ+1,κhas a left-inverse, thenf = 0. This shows thatχκ is a bijective

operator.

Let κ > 0, we define the dual intertwining operator ^tχκ on D(R) (D(R) is the space of C^∞-functions onR with compact support) by

tχ_κf

(x) = 1 Γ(κ)

Z +∞

|x|

(t− |x|)^κ−1t^−κf(sgn(x)t) dt, x∈R\ {0}.

Proposition 3. The operator ^tχκ is a topological automorphism of D(R), and satisfies the transmutation relation:

Z

R

(χ_κf)(x)g(x) dx= Z

R

f(x) ^tχ_κg

(x) dx, f ∈C^∞(R).

Proof . Letf ∈C^∞(R) andg∈ D(R), we have Z

R

(χκf)(x)g(x) dx= 1 Γ(κ)

Z +∞

0

Z x

0

(x−t)^κ−1f(t) dtg(x)x^−κdx

− 1 Γ(κ)

Z ∞ 0

Z x

0

(x−t)^κ−1f(−t) dtg(−x)x^−κdx.

Using Fubini’s theorem and a change of variable, we get Z

R

(χκf)(x)g(x) dx= 1 Γ(κ)

Z +∞

0

Z ∞ t

x^−κ(x−t)^κ−1g(x) dxf(t) dt

+ 1

Γ(κ) Z 0

−∞

Z ∞

−t

x^−κ(x+t)^κ−1g(−x) dxf(t) dt.

Therefore, Z

R

(χκf)(x)g(x) dx= 1 Γ(κ)

Z

R

Z ∞

|t|

x^−κ(x− |t|)^κ−1g(sign(t)x) dxf(t) dt

= Z

R

f(t) ^tχκg

(t) dt.

Proposition 4. Let κ >0, the Kummer transform F_κ satisfies the decomposition F_κ(f) =F ◦^tχ_κ(f), f ∈ D(R).

Proof . The result follows from Proposition 3.

6 Multivariable case

6.1 Direct product setting

In this subsection, we consider the direct product of the one-dimensional models, which means that the Weyl group of the corresponding subsystem of root system is a subgroup ofZ^N₂ .

We denote by τ_k (for each k from 1 to N) the orthogonal projection with respect to the hyperplane orthogonal toe_k, that is to say for everyx= (x₁, . . . , x_N)∈R^N

τ_k(x) =x−hx, e_ki

|e_k|² e_k= (x1, . . . , xk−1,0, x_k+1, . . . , xN).

Let κ = (κ₁, κ₂, . . . , κ_N) ∈ C^N. The associated Dunkl type operators T_j for j = 1, . . . , N, are given for x∈R^N by

T_jf(x) =∂_jf(x) +

N

X

l=1

κ_lf(x)−f(τ_l(x))

hx, e_li he_k, e_li

=∂jf(x) +κj

f(x)−f(x1, . . . , xj−1,0, xj+1, . . . , xN)

x_j .

These operators form a commuting system. The generalized Laplacian associated with T_j is defined in a natural way as

∆κ=

N

X

j=1

T_j².

A straightforward computation yields

∆κ= ∆ + 2

N

X

j=1

κjx⁻¹_j ∂jf(x)−

N

X

j=1

(κ²_j +κj)x⁻¹_j ∂jf(x1, . . . , xj−1,0, xj+1, . . . , xN)

+

N

X

j=1

(κ²_j−κj)x⁻²_j f(x)−f(x1, . . . , xj−1,0, xj+1, . . . , xN) .

This operator will play in our context a similar role to that of the Euclidean Laplacian in the classical harmonic analysis. Obviously, the trivial choice of the multiplicity functionκ= 0, reduces our situation to the analysis related to the classical Laplacian ∆.

Letκ = (κ₁, . . . , κ_N) ∈(0,∞)^N. For x, λ∈R^N, we consider the function M_κ(λ, x) which is given as the tensor products

Mκ(λ, x) =

N

Y

j=1

Mκj(iλjxj).

Theorem 6. Forλ= (λ₁, . . . , λ_N)∈C^N, the function M_κ(λ, x) is the unique analytic solution of the system

T_ξu(x) =ihλ, ξiu(x), u(0) = 1, ∀ξ∈C^N.

6.2 Dunkl-type operators associated to an orthogonal subsystem in a root system of type AN−1

Let R be a root system of typeAN−1

R={±(e_i−e_j),1≤i < j ≤N}.

Define a positive orthogonal subsystem R⁰ ={α₁, . . . , α_[N/2]}of R by setting:

α_i=e2i−1−e_2i, i= 1, . . . ,[N/2].

We denote by τ_j (for each j from 1 to [N/2]) the orthogonal projection onto the hyperplane perpendicular to α_j, that is to say for everyx= (x₁, . . . , x_N)∈R^N

τ_ix= (x₁, . . . , x2i−1, x_2i, . . . , x_N),

where x2i−1 = x2i = ¹₂(x2i−1 +x2i), i= 1, . . . ,[N/2]. The vector ξ ∈ R^N can be decomposed uniquely in the form

ξ =

[N/2]

X

i=1

ξi(e2i−1−e2i) +ξ,b

where ξbis an orthogonal vector to the linear space generated by R⁰ ={α₁, . . . , α_[N/2]}.

A straightforward computation shows that the operatorT_ξ (ξ∈R^N) associated withR⁰ and the multiplicity parameters (κ₁, . . . , κ_[N/2]) has the following decomposition

T_ξ=

[N/2]

X

i=1

ξ_iT_αi+∂

ξb=

2[N/2]

X

i=1

(−1)ⁱ⁺¹ξ_[i+1 2 ]T_i+∂

ξb, where

T_i=∂_i−(−1)ⁱκ_[i+1 2 ]ρ_[i+1

2 ], i= 1, . . . ,2[N/2], and

(ρ_if)(x) = f(x)−f(τ_ix) x2i−1−x2i

.

The intertwining operator (3) becomes χ_κ(f)(x) = 1

Γ(κ) Z

[0,1]ⁿ

f(h(t, x))w(t) dt, where

h(t, x) =x+

[N/2]

X

i=1

t_i−1

2 (x2i−1−x2i)(e2i−1−e2i).

Proposition 5. Let λ = (λ1, . . . , λ_N) ∈ C^N, and κ = (κ1, . . . , κ_[N/2]) ∈ (0,∞)^[N/2]. The following eigenvalue problem

T_ξf =ihλ, ξif, f(0) = 1, ∀ξ∈C^N, (11)

has a unique analytic solution M_κ(λ, x) given by M_κ(λ, x) =eihλ,h(0,x)i

[N/2]

Y

j=1

M_κj i

2(λ2j−1−λ_2j)(x2j−1−x_2j)

.

Proof . According to Theorem 2, χ_κ is an intertwining operator between T_ξ and ∂_ξ. So, the functionχκ(e^ihλ,·i) is the unique C^∞-solution of problem (11).

Since we can write

hλ, h(t, x)i=hλ, h(0, x)i+

[N/2]

X

j=1

tj

2(λ2j−1−λ2j)(x2j−1−x2j), we are lead to

Mκ(λ, x) = eihλ,h(0,x)i

Γ(κ) Z

[0,1]^[N/2]

e

i 2

[N/2]

P

j=1

tj(λ2j−1−λ_2j)(x2j−1−x_2j)

w(t) dt

=eihλ,h(0,x)i [N/2]

Y

j=1

1 Γ(κj)

Z

[0,1]

e^{i2(λ2j−1−λ2j)(x2j−1−x2j)}(1−t_j)^κj−1dt_j. If we now use (7) and (8) we get

M_κ(λ, x) =eihλ,h(0,x)i [N/2]

Y

j=1

M_κj i

2(λ2j−1−λ_2j)(x2j−1−x_2j)

.

6.3 Dunkl-type operators associated to orthogonal subsystem in root system of type B_N

Throughout this subsectionR is a root system of type BN which is given by R={±e_i±e_j,1≤i < j ≤N; ±e_i1≤i≤N},

and R⁰ is a positive orthogonal subsystem R⁰ in the root systemR given by R⁰={α^±_i =e2i−1±e_2i,1≤i≤[N/2]}.

Denote by τ_i^± (for each i from 1 to [N/2]) the orthogonal projection onto the hyperplane per- pendicular to α^±_i , that is to say for everyx= (x1, . . . , xN)∈R^N

τ_i^±x= x₁, . . . , x^±_2i−1, x^±_2i, . . . , x_N ,

wherex^±_2i−1 =x^±_2i= ¹₂(x2i−1±x2i). In this case, the Dunkl type operatorT_ξassociated with R⁰ and the multiplicity parameters κ^±₁, . . . , κ^±_[N/2]

takes the form (T_ξf)(x) =∂_ξf(x) +

[N/2]

X

j=1

κ⁻_j hα⁻_j , ξif(x)−f(τ_j⁻x)

hx, α⁻_j i +κ⁺_jhα⁺_j , ξif(x)−f(τ_j⁺x) hx, α⁺_j i . In particular, for i= 1, . . . ,2[N/2] we have

T_i=∂_i−(−1)ⁱκ⁻

[ⁱ⁺¹₂ ]ρ⁻

[ⁱ⁺¹₂ ]+κ⁺

[ⁱ⁺¹₂ ]ρ⁺

[ⁱ⁺¹₂ ]. where

(ρ^±_i f)(x) = f(x)−f(τ_i^±x) x2i−1±x_2i .

The operator T_ξ has also the following decomposition T_ξ=

2[N/2]

X

i=1

ξ⁺

[ⁱ⁺¹₂ ]+ (−1)ⁱ⁺¹ξ⁻

[ⁱ⁺¹₂ ]

T_i+εξ_N∂_N, where

ξ =

[N/2]

X

i=1

ξ_i⁺α⁺_i +ξ⁻_i α⁻_i +εξNeN, and ε=

(1 ifN is odd, 0 ifN is even.

Proposition 6. Let λ= (λ1, . . . , λN)∈C^N and κ= (κ⁺, . . . , κ⁺_[N/2], κ⁻, . . . , κ⁻_[N/2])∈(0,∞)^2[N/2]. The following eigenvalue problem

T_ξf =ihλ, ξif, f(0) = 1, ∀ξ∈C^N, has a unique analytic Mκ(λ, x) given by

M_κ(λ, x) =eihλ,h(0,x)i [N/2]

Y

j=1

M_κ⁻

j

i

2(λ2j−1−λ_2j)(x2j−1−x_2j)

×M_κ+ j

i

2(λ2j−1+λ_2j)(x2j−1+x_2j)

.

Acknowledgements

This research is supported by NPST Program of King Saud University, project number 10- MAT1293-02. I would like to thank the editor and the anonymous referees for their helpful comments and remarks.

References

[1] Bouzeffour F., Special functions associated with complex reflection groups,Ramanujan J., to appear.

[2] Cherednik I., Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and Macdonald’s operators, Int. Math. Res. Not.(1992), 171–180.

[3] Dunkl C.F., Reflection groups and orthogonal polynomials on the sphere,Math. Z.197(1988), 33–60.

[4] Dunkl C.F., Differential-difference operators associated to reflection groups,Trans. Amer. Math. Soc.311 (1989), 167–183.

[5] Dunkl C.F., Opdam E.M., Dunkl operators for complex reflection groups, Proc. London Math. Soc. 86 (2003), 70–108,math.RT/0108185.

[6] Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.

[7] Heckman G.J., An elementary approach to the hypergeometric shift operators of Opdam,Invent. Math.103 (1991), 341–350.

[8] Heckman G.J., Dunkl operators,Ast´erisque 245(1997), Exp. No. 828, 4, 223–246.

[9] Humphreys J.E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.

[10] Kober H., On fractional integrals and derivatives,Quart. J. Math., Oxford Ser.11(1940), 193–211.

[11] Koornwinder T.H., Bouzeffour F., Nonsymmetric Askey–Wilson polynomials as vector-valued polynomials, Appl. Anal.90(2011), 731–746,arXiv:1006.1140.

[12] Luchko Y., Trujillo J.J., Caputo-type modification of the Erd´elyi–Kober fractional derivative,Fract. Calc.

Appl. Anal.10(2007), 249–267.

[13] Macdonald I.G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.

[14] Opdam E.M., Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group,Compositio Math.85(1993), 333–373.

[15] Temme N.M., Special functions. An introduction to the classical functions of mathematical physics,A Wiley- Interscience Publication, John Wiley & Sons Inc., New York, 1996.