# Dunkl-Poisson Equation and Related Equations in Superspace.

1 Introduction

Dunkl operators (differential-difference operators), introduced by Dunkl [9], are invariant under a finite reflection group [member of] and are also pairwise commuting. One of the interesting aspects of these operators is that they allow for the construction of a Dunkl Laplace operator, which is a combination of the classical Laplace operator in [R.sup.m] with some difference terms, such that the resulting operator is only invariant under [member of] and not under the whole orthogonal group. Moreover, these operators not only provide a useful tool in the study of special functions with root systems [10], but they are also closely related to affine Hecke algebras [26] and integrable system of Calogero-Moser-Sutherland type [13].

One of the main aims of Clifford analysis is to study the function-theoretical properties of the null-solutions of the Dirac operator which is invariant under rotations but not under reflections [7]. A Dunkl version of the Dirac operator in Clifford analysis introduced by Cerejeiras et al, is invariant under reflection groups and also factorizes the Dunkl-Laplacian [8]. Then, they obtained a Stokes theorem, a Borel-Pompeiu formula and a Cauchy integral formula for the Dirac operator, and investigated a Fueter's theorem and Fischer decompositions in Dunkl Clifford analysis(see [3,11,12]). More recently, Sommen, DeBie and others studied Clifford analysis in superspace [R.sup.m]|2n (see [4,5]). Superspaces are spaces equipped with both a set of commuting variables and a set of anticommuting variables in order to describe the properties of bosons and fermions in Quantum Mechanics. In 2010, Ren gave the Fischer decomposition on the space of spinor valued polynomials in the superspace for the super DunklDirac operator with the bosonic Dunkl-Dirac operator (i.e., the Dunkl-Dirac operator in [R.sup.m]) and the fermionic Dirac operator(see [24]). Based on the above-mentioned results, we investigated the Almansi type expansion for super Dunkl-Laplace operators.

In 1899, the Almansi expansion for polyharmonic functions was established [1]. Indeed the expansion builds the relation between harmonic functions and polyharmonic functions, which plays a central role in the theory of polyharmonic functions. The result in the case of harmonic analysis, complex analysis, Clifford analysis, and Clifford analysis in superspace have been well developed in [2,19,21,23,25,28]. In the present paper, we study Almansi type expansions for solutions of Dunkl-polyharmonic equations in superspace by normalized systems.

Normalized systems of functions were advocated by Bondarenko [6]. Afterwards, Karachik constructed 0-normalized system of functions with respect to a Laplace operator and applied the system to an expansion of Almansi type for polyharmonic functions in [R.sup.m] (see [14, 15, 16]). Then, normalized systems for wave operators, Dunkl operators, super Dirac operators are obtained (see [20,22,27]). In particular, applying normalized systems and Almansi expansions, Karachik studied solutions of some partial equations and some boundary value problems for Poisson's Equation (see [17,18]). But as far as we know, up to now there is no hint on normalized systems in Dunkl superspace. In this paper, we try to fill part of this gap by studying solutions to Dunkl-Helmholtz equations, Dunkl-Poisson equations, inhomogeneous Dunkl-polyharmonic equations in superspace using normalized systems.

2 Preliminaries

2.1 Dunkl-Clifford analysis in [R.sup.m]

Denote by (.,.} the standard Euclidean scalar product in [R.sup.m] and by [absolute value of x] = [(x,x).sup.1/2] associated norm. For [alpha] [member of] [R.sup.m]\{0} the reflection [[sigma.sub.[alpha]] in the hyperplane orthogonal to a is given by

[[sigma].sub.[alpha]] (x) = x - 2 ([alpha], x)/[[absolute value of [alpha]].sup.2] [alpha], x [member of] [R.sup.m].

A finite set R [subset] [R.sup.m] \ {0} is called a root system if [alpha]R [intersection] R = {[alpha], -[alpha]} and [[sigma].sub.[alpha]]R = R for all [alpha] [member of] R. For a given root system R, the reflections [[sigma].sub.[alpha]], [alpha] [member of] R generate the finite group [member of] [subset] O(m), called the finite reflection group (or the Coxeter group) associated with R.

A multiplicity function [kappa] on the root system R is a G-invariant function [kappa] : R [right arrow] C i.e. [kappa]([alpha]) = [kappa](h[alpha]) for all h [member of] G. We will denote [kappa]([alpha]) by [[kappa].sub.[alpha]].

For each fixed positive subsystem R+ and multiplicity function k we have the Dunkl operators (also, differential-difference operators):

[mathematical expression not reproducible],

for f [member of] [C.sup.1] ([R.sup.m]). An important consequence is that the operators [T.sub.i] are mutually commutating, that is, [T.sub.i][T.sub.j] = [T.sub.j][T.sub.i]. The Dunkl-Laplace operator is given by

[mathematical expression not reproducible]

with the classical Laplace operator [DELTA] and the gradient operator [nabla].

We notice that [mathematical expression not reproducible], and the Dunkl dimension [mu] = m + 2[gamma]. Henceforward, we assume [kappa] [greater than or equal to] 0 and [[gamma].sub.k] > 0.

We consider functions f : [R.sup.m] [right arrow] [R.sub.0,m]. Hereby [R.sub.0,m] denotes the Clifford algebra over [R.sup.m] generated by {[e.sub.1], [e.sub.2], ..., [e.sub.m]} satisfying the anti-commutation relationship [e.sub.i] [e.sub.j] + [e.sub.j] [e.sub.i] = - 2[[delta].sub.ij], where [[delta].sub.ij] is the Kronecker symbol. [x.bar] = [m.summation over (i=1)] [x.sub.i] [e.sub.i] is the so-called vector variable.

A Dunkl-Dirac operator in [R.sup.m] for the corresponding reflection group [member of] is defined as [D.sub.h] f = [m.summation over (i=1)] [e.sub.i] [T.sub.i] f. Functions belonging to the kernel of the Dunkl-Dirac operator [D.sub.h] are called Dunkl-monogenic functions. Moreover, - [D.sup.2.sub.h] = [[DELTA].sub.h], where [[DELTA].sub.h] is called the Dunkl-Laplace operator in [R.sup.m]. Functions belonging to the kernel of Dunkl-Laplace operator are called Dunkl-harmonic functions.

2.2 Dunkl-Clifford analysis in [R.sup.m|2n]

On a superspace of dimension (m, 2n), we have m commuting (or bosonic) variables [x.sub.1], ..., [x.sub.m] and 2n anti-commuting (or fermionic) variables [mathematical expression not reproducible] subject to

Furthermore we know the Clifford algebra generators [e.sub.1],..., [e.sub.m] and the symplectic Clifford algebra generators [mathematical expression not reproducible]. They obey the following rules:

[mathematical expression not reproducible],

Taking the above relations into account, we study the superspace by the real algebra:

[mathematical expression not reproducible],

which is nothing than the tensor product of [mathematical expression not reproducible]. The algebra Alg([x.sub.i]; [[??].sub.j]) is called a scalar algebra denoted by V and the algebra Alg([e.sub.i], [[??].sub.j]) is a Clifford algebra denoted by [C.sub.m|2n]. Moreover, the elements of both two algebras can commute with each other. When n = 0, we have that [mathematical expression not reproducible] generated by the commuting variables [x.sub.i]. In the case [C.sub.m|0] [congruent to] [R.sub.0,m], [R.sub.0,m] is the standard orthogonal Clifford algebra. When m = 0, we have that V [cross product] [C.sub.0|2n] = [[LAMBDA].sub.2n] [cross product] [W.sub.2n], with [[LAMBDA].sub.2n] being the Grassmann algebra generated by [[??].sub.j]. In the case [C.sub.0|2n] [congruent to] [W.sub.2n], [W.sub.2n] is the Weyl algebra generated by [[??].sub.j].

We define the super vector variable [mathematical expression not reproducible]. By direct calculation, we obtain the square of x:

[mathematical expression not reproducible].

Note that [[x.bar].sup.2] = - [m.summation over (i=1)] [x.sup.2.sub.i] is the minus norm squared of a vector in Euclidean space.

Finally, we define a more general function space as [C.sup.k]([OMEGA]) [cross product] [[LAMBDA].sub.2n] [cross product][C.sub.m|2n], where [C.sup.k] ([OMEGA]) denotes the space of the k-times continously differentiables real-valued functions defined in some domain [OMEGA] [subset] [R.sup.m]. We use the notation

[C.sup.k] [([OMEGA]).sub.m|2n] = [C.sup.k] ([OMEGA]) [cross product] [[LAMBDA].sub.2n].

The super Dunkl-Dirac operator is defined to be

[mathematical expression not reproducible]

with [D.sub.h] and [D.sub.f] the bosonic Dunkl-Dirac operator and the fermionic Dirac operator, respectively.

If we let D act on x, we see that

M := 1/2 Dx = -n + -n + m/2 + [[gamma].sub.k],

where M is the so-called super Dunkl dimension in contrast to the non-Dunkl case of the super-dimension m - 2n in [5]. The numerical parameter M is regarded as the ground level energy in physics.

As usual, functions belonging to the kernel of the super Dunkl-Dirac operator are called super Dunkl-monogenic functions.

The super Dunkl-Laplace operator is the square of the super Dunkl-Dirac operator

[mathematical expression not reproducible].

The Dunkl-Laplace operator [[DELTA].sub.h] is invariant under the Coxeter group G, while the fermionic Laplace operator [[DELTA].sub.f] is invariant under the symplectic group. Besides, we define the super Euler operator as

[mathematical expression not reproducible].

It is easy to decompose P as

[mathematical expression not reproducible].

3 Normalized system for the super Dunkl-Laplace operator

Definition 1. [2] An open connected set [OMEGA] [subset] [R.sup.m] is a star domain with center 0 if any [x.bar] [member of] [OMEGA] and 0 [less than or equal to] t [less than or equal to] 1 imply that t[x.bar] [member of] [OMEGA]. The set is denoted by [[OMEGEA].sup.*].

Definition 2. Let f (x) [member of] [C.sup.1] [([[OMEGA].sup.*]).sub.m|2n] [cross product] [C.sub.m|2n]. The operator [J.sub.s] is defined as

[mathematical expression not reproducible],

where s > 0 and M > 0.

Definition 3. We define the operator [E.sub.t] by

[mathematical expression not reproducible],

where t [member of] R and I is the identity operator.

Lemma 1. [24] The operators [chi square], A, E have the following properties:

[DELTA][chi square] - [chi square][DELTA] = 4[E.sub.M], [E.sub.M][chi square] - [chi square] [E.sub.M] = 2[chi square],

where M = -n + m/2] + [[gamma].sub.[kappa]].

Lemma 2. Let [mathematical expression not reproducible], then

[mathematical expression not reproducible].

Proof. By Lemma 1, we have

[mathematical expression not reproducible].

Thus, we have the proof.

Lemma 3. Let f (x) [member of] [C.sup.1] [([[OMEGA].sup.*]).sub.m|2n] [cross product] [C.sub.m|2n]. If s > 1, then

[E.sub.M+s-1] [J.sub.s] f(x) = (s - 1) [J.sub.s-1] f(x).

Proof. We calculate

[mathematical expression not reproducible].

Let f ([alpha]x) = [f.sub.1]([alpha][x.bar]) [f.sub.2]([alpha][[??].bar]). Applying integration by parts, we have

[mathematical expression not reproducible].

To sum up, we have the conclusion.

Theorem 1. Let f (x) [member of] [C.sup.2] [([[OMEGA].sup.*]).sub.m|2n] [cross product] [C.sub.m|2n]. If [DELTA]f (x) = 0, then the system

[mathematical expression not reproducible]

is the 0-normalized system for the super Dunkl-Laplace operator, where [J.sub.s]f is given in Definition 2.

Proof. Note that [DELTA][J.sub.s]f = 0 for s [greater than or equal to] 1. Lemmas 2 and 3 imply that

[mathematical expression not reproducible].

Therefore, we obtain the conclusion.

4 Applications of the normalized system

4.1 Almansi type expansion for solutions of Dunkl-polyharmonic equations in superspace

Lemma 4. For q [member of] N and l [greater than or equal to] 0,

[mathematical expression not reproducible]. (4.1)

For q = 0,

(E + l + 1) [[integral].sup.1.sub.0] [[alpha].sup.l]g([alpha]x)d[alpha] = g(x). (4.2)

Proof. For q = 0, we calculate

[mathematical expression not reproducible].

Hence we see that

(E + l + 1) [[integral].sup.1.sub.0] [[alpha].sup.l]g([alpha]x)d[alpha] = g(x).

Then it is easy to obtain (4.1). Note that (4.1) is (4.2) for the case q = 0. Similarly, for q [member of] N and l [greater than or equal to] 0, we have

[mathematical expression not reproducible]

which completes the proof.

Theorem 2. Let G(x) [member of] [C.sup.2r] [([[OMEGA].sup.*]).sub.m|2n] [cross product] [C.sub.m|2n]. If [[DELTA].sup.r] tG(x) = 0, then

[mathematical expression not reproducible], (4.3)

where [DELTA] [f.sub.s] (x) = 0 (0 [less than or equal to] s [less than or equal to] r - 1), and

[mathematical expression not reproducible]. (4.4)

Proof. First we will prove that [f.sub.s] s(x) in (4.4) satisfy (4.3). Inserting (4.4) into the right-hand side of (4.3), we have

[mathematical expression not reproducible]. (4.5)

Denote by [A.sub.1](x) the fourth term on the right side of the (4.5). Then

[mathematical expression not reproducible].

Denote by [A.sub.2](x) the integral in the above expression. Let t = [alpha][beta], then dt = [alpha]d[beta] for 0 [less than or equal to] [alpha] [less than or equal to] 1 and 0 [less than or equal to] t [less than or equal to] [alpha]. We calculate

[mathematical expression not reproducible].

Let [alpha] = [beta] + t. Then d[beta] = d[alpha]. For 0 [less than or equal to] [beta] [less than or equal to] 1 - t,

[mathematical expression not reproducible].

Let [beta] = [alpha](1 -1). Then d[beta] = (1 - t)d[alpha] for 0 [less than or equal to] [alpha] [less than or equal to] 1. We see that

[mathematical expression not reproducible].

We calculate

[mathematical expression not reproducible],

where

B(l, s) = [[integral].sup.1.sub.0] [[alpha].sup.l-1] ([1 - [alpha]).sup.s-1] d[alpha].

Using the relation between Beta functions and Gamma functions:

B(l, s) = T(l)[GAMMA](s)/[GAMMA](s + l), [GAMMA](s) = (s - 1)!,

we can write

[mathematical expression not reproducible].

Inserting [A.sub.3](t) into [A.sub.1](x), we have

[mathematical expression not reproducible].

We calculate

[mathematical expression not reproducible],

and

[mathematical expression not reproducible].

Thus, we have

[mathematical expression not reproducible].

Inserting [A.sub.1](x) into (4.5), we have (4.3).

Next, we will prove that [DELTA][f.sub.s](x) = 0. By Lemma 2, we have

[mathematical expression not reproducible].

By means of Lemma 4, we have

[mathematical expression not reproducible].

As suggested above, we get

[DELTA][f.sub.s](x) = 0.

Therefore, we have the conclusion.

4.2 Solutions of the Dunkl-Helmholtz equation in superspace

In this subsection, we investigate the solutions of the Dunkl-Helmholz equation in superspace

([DELTA] + [lambda]) G (x) = 0, (4.6)

where the constant [lambda] [member of] C, G(x) [member of] [C.sup.1] [([[OMEGA].sup.*]).sub.m|2n] [cross product] [C.sub.m|2n].

Theorem 3. Let f (x) [member of] [C.sup.[infinity]][([[OMEGA].sup.*]).sub.m|2n] [cross product] [C.sub.m|2n]. Suppose that the function f (x) is super Dunkl-harmonic and M > 0. Then

G(x) = [[infinity].summation over (s=0)] [(-[lambda]).sup.s] [F.sub.s](x; f) (4.7)

is a formal solution for the equation (4.6), where [F.sub.s](x; f) are given in Theorem 1.

Proof. By Theorem 1, we have

[mathematical expression not reproducible].

So the series G(x) satisfies formally equation (4.6).

4.3 Solutions of the Dunkl-Poisson equation and the inhomogeneous Dunkl-polyharmonic equation in superspace

Consider the Dunkl-Poisson equation in superspace

[DELTA]g = f (x), (4.8)

where f (x) [member of] [C.sup.[infinity]][([[OMEGA].sup.*]).sub.m|2n] [cross product] [C.sub.m|2n].

Theorem 4. Let f (x) [member of] [C.sup.[infinity]][([[OMEGA].sup.*]).sub.m|2n] [cross product] [C.sub.m|2n]. Then the function G(x) given by

[mathematical expression not reproducible] (4.9)

is a formal solution of the equation (4.8).

Proof. By Lemmas 2 and 4, we have

[mathematical expression not reproducible].

Thus, we complete the proof.

Consider the inhomogeneous Dunkl-polyharmonic equation in superspace

[[DELTA].sup.k] g = f (x), (4.10)

where f (x) [member of] [C.sup.[infinity]] [([[OMEGA].sup.*]).sub.m|2n] [cross product] [C.sub.m|2n]

Applying Theorem 4, we can obtain the following theorem by induction.

Theorem 5. Let f (x) [member of] [C.sup.[infinity]][([[OMEGA].sup.*]).sub.m|2n] [cross product] [C.sub.m|2n]. Then the function G(x) given by

[mathematical expression not reproducible] (4.11)

is a formal solution of the equation (4.10).

Remark 1. If the series G(x) in (4.7) converges absolutely and uniformly, then it is a classical solution of the equation (4.6). Similarly, the series G(x) in (4.9) and (4.11) can be considered.

http://dx.doi.org/10.3846/13926292.2015.1112856

Acknowledgements

This research was supported by the TianYuan Special Funds of the National Natural Science Foundation of China (No. 11426082).

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Hong Fen Yuan (a) and Valery V. Karachik (b)

(a) Hebei University of Engineering 056038 Handan, China

(b) South Ural State University 454080 Chelyabinsk, Russia

E-mail(corresp.): yuanhongfen1980@126.com

E-mail: Karachik@susu.ru

Received March 13, 2015; revised September 30, 2015; published online November 15, 2015
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