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Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator.

1 Introduction

Dunkl operators (see [9, 10]) are combinations of differential and difference operators, associated to a finite reflection group G. One of the interesting aspects of these operators is that they allow for the construction of a Dunkl Laplacian, which is a combination of the classical Laplacian in [R.sup.m] with some difference terms, such that the resulting operator is only invariant under G and not under the whole orthogonal group. Moreover, they are directly related to quantum integrable models of Calogero type (see e.g. [13]) and have as such received a lot of attention in the physics literature.

Clifford analysis (see [1, 7]), in its most basic form, is a refinement of the theory of harmonic analysis in m-dimensional Euclidean space. By introducing the so-called Dirac operator, the square of which equals the Laplace operator, one introduces the notion of monogenic functions. These are, at the same time, a refinement of harmonic functions and a generalization of holomorphic functions in one complex variable.

Generalizations of the classical Gegenbauer polynomials to the Clifford analysis framework are called Clifford-Gegenbauer polynomials and were introduced as well on the closed unit ball B(1) (see [4]), as on the Euclidean space [R.sup.m] (see [2,8]). For the superspace case, which can be seen as the study of differential operators invariant under the action of the group O(m) x Sp(2n), we refer to [5]. In this paper, we adapt the definition of the Clifford-Gegenbauer polynomials, both on B(1), as on Rm, to the case of Dunkl operators.

The paper is organized as follows. In Section 2 we first give some background on Dunkl operators. Then we prove some fundamental results concerning the Dunkl Dirac operator and its nullsolutions, called Dunkl monogenics. In Section 3 we introduce Clifford-Gegenbauer polynomials on B(1) related to the Dunkl Dirac operator. Basic properties, such as a Rodrigues formula, a differential equation, recurrence relations and an orthogonality relation are derived. Moreover, we obtain an expression of these newly introduced polynomials in terms of the Jacobi polynomials on the real line. Next, the Clifford-Gegenbauer polynomials on [R.sup.m] are adapted to the case of Dunkl operators (Section 4). They satisfy similar properties as their counterparts on the unit ball. However, the orthogonality must be treated completely different; it is expressed in terms of a bilinear form instead of an integral.

2 Clifford Dunkl setting

2.1 Dunkl operators

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

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

A root system is a finite subset R [subset] [R.sup.m] of non-zero vectors such that, for every [alpha] [member of] R, the associated reflection [r.sub.[alpha]] preserves R. We will assume that R is reduced, i.e. R [intersection] R[alpha] = {[+ or -][alpha]} for all [alpha] [member of] R. Each root system can be written as a disjoint union R = [R.sub.+] [union] (-[R.sub.+]), where [R.sub.+] and -[R.sub.+] are separated by a hyperplane through the origin. The subgroup G [subset] O(m) generated by the reflections {[r.sub.[alpha]] | [alpha] [member of] R} is called the finite reflection group associated with R. We will also assume that R is normalized such that <[alpha], [alpha]> = 2 for all [alpha] [member of] R. For more information on finite reflection groups we refer the reader to [12].

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

Fixing a positive subsystem [R.sub.+] of the root system R and a multiplicity function k, we introduce the Dunkl operators [T.sub.i] associated to [R.sub.+] and k by (see [9,10])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

An important property of the Dunkl operators is that they commute, i.e. [T.sub.i][T.sub.j] = [T.sub.j][T.sub.i]

The Dunkl Laplacian is given by [[DELTA].sub.k] = [[summation].sup.m.sub.i=1] [T.sup.2.sub.i], or more explicitly by

[[DELTA].sub.k]f(x) = [DELTA]f(x) + [summation over ([alpha] [member of] [R.sub.+])] k[alpha] (<[nabla]f(x), [alpha]>/<[alpha], x> - f(x) - f([r.sub.[alpha]](x))/[<[alpha], x>.sup.2])

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

If we let [[DELTA].sub.k] act on [[absolute value of x].sup.2] we find [[DELTA].sub.k][[[absolute value of x].sup.2]] = 2m + 4[gamma] = 2[mu], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We call [mu] the Dunkl dimension, because most special functions related to [[DELTA].sub.k] behave as if one would be working with the classical Laplace operator in a space with dimension [mu].

The operators

E := 1/2 [[absolute value of x].sup.2], F := -1/2 [[DELTA].sub.k] and H := E + [mu]/2

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Euler operator, satisfy the defining relations of the Lie algebra [sl.sub.2] (see e.g. [11]). They are given by

[H, E] = 2E, [H, F] = -2F, [E, F] = H. (1)

We also have the following important property of the Dunkl operators

[T.sub.i][fg] = [T.sub.i][f] g + f [T.sub.i][g] , i = 1,2, ..., m (2)

if f or g are G-invariant.

2.2 Dunkl Dirac operators

For the sequel of this paper we restrict ourselves to multiplicity functions satisfying [k.sub.[alpha]] [greater than or equal to] [for all][alpha] [member of] [R.sub.+]; hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and for the Dunkl dimension [mu] we have that [mu] = m + 2[gamma] > 1.

From now on we consider functions f : [R.sub.m] [right arrow] [R.sub.0,m]. Hereby, [R.sub.0,m] denotes the Clifford algebra over [R.sup.m] generated by [e.sub.i], i = 1, ..., m, subject to the relations [e.sub.i][e.sub.j] + [e.sub.j][e.sub.i] = -2[[delta].sub.ij].

In what follows, the bar denotes the Clifford conjugation; an anti-involution for which [bar.[e.sub.i]] = -[e.sub.i] (i = 1, ..., m).

[D.sub.k] stands for the Dunkl Dirac operator [D.sub.k] = [[summation].sup.m.sub.j=1] [e.sub.j][T.sub.j] with [T.sub.j] the Dunkl operators, [x.bar] = [[summation].sup.m.sub.j=1] [e.sub.j][x.sub.j] is the so-called vector variable. It is clear that [D.sup.2.sub.k] = -[[DELTA].sub.k] and that [[x.sup.2].bar] = -[[absolute value of [x.bar]].sup.2] = -[r.sup.2].

By definition we have that for f([x.bar]) = [[summation].sub.A][f.sub.A]([x.bar]) with [f.sub.A] : [R.sup.m] [right arrow] R :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A function f which satisfies [D.sub.k][f] = 0 is called left Dunkl monogenic, while a function f which satisfies [f][D.sub.k] = 0 is called right Dunkl monogenic.

A left Dunkl monogenic homogeneous polynomial [M.sub.k] of degree k (k [greater than or equal to] 0) in [R.sup.m] is called a left solid inner Dunkl monogenic of order k. A left Dunkl monogenic homogeneous function [Q.sub.k] of degree -(k + [mu] - 1) in [R.sup.m] \ {0} is called a left solid outer Dunkl monogenic of order k. The restriction to [S.sup.m-1] of a left solid inner Dunkl monogenic is called a left inner Dunkl monogenic, while the restriction to [S.sup.m-1] of a left solid outer Dunkl monogenic is called a left outer Dunkl monogenic. The set of all left solid inner Dunkl monogenics of order k will be denoted by [M.sup.+.sub.l](k), while the set of all left solid outer Dunkl monogenics of order k will be denoted by [M.sup.-.sub.l](k). Moreover, the space of left inner Dunkl monogenics of order k is denoted by [M.sup.+.sub.l](k), while the space of left outer Dunkl monogenics is denoted by [M.sup.-.sub.l](k). Similar definitions hold in case of right Dunkl monogenics.

The angular Dunkl Dirac operator (or Gamma operator) is defined as [[GAMMA].sub.k] := [D.sub.k][x.bar] + [mu] + E.

Taking into account the relation (see for e.g. [6,15])

{[D.sub.k], [x.bar]} = [D.sub.k][x.bar] + [x.bar][D.sub.k] = -(2E + [mu]), (3)

we easily obtain the following proposition.

Proposition 2.1.

One has that

[x.bar][D.sub.k] = -E -[[GAMMA].sub.k] or [[GAMMA].sub.k] = -[x.bar][D.sub.k] - E.

Proof. We have consecutively

[x.bar][D.sub.k] = -[D.sub.k][x.bar] - 2E - [mu] = -[[gamma].sub.k] + [mu] + E - 2E - [mu] = -[[GAMMA].sub.k] - E.

Now we immediately obtain that E measures the degree of homogeneity, while [[GAMMA].sub.k] measures the degree of Dunkl monogenicity.

Proposition 2.2.

(i) [R.sub.k] [member of] [P.sub.k] : E[[R.sub.k]] = k[R.sub.k]

(ii) [M.sub.k] [member of] [M.sup.+.sub.l](k) : [[GAMMA].sub.k][[M.sub.k]] = -k[M.sub.k]

(iii) [M.sub.k] [member of] [M.sup.+.sub.l](k) : [[GAMMA].sub.k][[x.bar] [M.sub.k]] = (k + [mu] - 1) [x.bar][M.sub.k].

Proof.

(i) Straightforward.

(ii) [[GAMMA].sub.k][[M.sub.k]] = (-E - [x.bar][D.sub.k])[[M.sub.k]] = -k[M.sub.k].

(iii) By means of (i) and (ii) we obtain

[x.bar][D.sub.k][[x.bar]][M.sub.k]] = [x.bar] ([[GAMMA].sub.k] - [mu] - E) [[M.sub.k]] = [x.bar] (-k - [mu] - k) [M.sub.k] = (-2k - [mu]) [x.bar][M.sub.k].

Using Proposition 2.1 and the fact that [x.bar][M.sub.k] [member of] [P.sub.k+1], gives

(-2k - [mu]) [x.bar][M.sub.k] = (-E - [[GAMMA].sub.k])[[x.bar][M.sub.k]] = -(k + 1) [x.bar][M.sub.k] - [[GAMMA].sub.k][[x.bar][M.sub.k]],

which yields the desired result.

The following lemma is proved by induction using (3).

Lemma 2.1.

For [M.sub.k] [member of] [M.sup.+.sub.l] (k) we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now the following connection between left solid inner and outer Dunkl monogenics can be proved.

Proposition 2.3.

(i) For [M.sub.k] [member of] [M.sup.+.sub.l] (k) we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(ii) For [[??].sub.k] [member of] [M.sup.-.sub.l](k) we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence on the unit sphere [S.sup.m-1] we have the following connection

[M.sub.k] [member of] [M.sup.+.sub.l](k) [??] [[omega].bar]][M.sub.k]([[omega].bar]]) [member of] [M.sup.-.sub.l](k), [[omega].bar]] [member of] [S.sup.m-1].

Proof

(i) Clearly [Q.sub.k] is homogeneous of degree -(k + [mu] - 1). Moreover taking into account the product rule (2), we obtain that in [R.sup.m] \ {0} :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(ii) The converse result is proved in a similar way.

A straightforward calculation yields the following lemma.

Lemma 2.2.

One has [D.sub.k][[x.bar]] = -[mu].

Proof. From (3) we obtain that {[D.sub.k], [x.bar]}[l] = -(2E + [mu])[1] and hence [D.sub.k][[x.bar]] = -[mu].

The angular Dunkl Dirac operator only acts on the angular co-ordinates, whence its name.

Proposition 2.4.

One has that [[[gamma].sub.k], f(r)] = 0.

Proof. Let G([x.bar]) denote a Clifford algebra-valued function. Taking into account Proposition 2.1 and the product rule (2), we arrive at

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which proves the statement.

We also have the following important decomposition (see [15]).

Theorem 2.1 (Fischer decomposition). The space [P.sub.k] of homogeneous polynomials of degree k taking values in [R.sub.0,m] decomposes as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [M.sup.+.sub.l]{k - i) is the space of left solid inner Dunkl monogenics of order k - i.

Using the Gamma operator, it is now possible to construct projection operators on each piece in the Fischer decomposition in a much easier way than was obtained in [15]. Indeed, it is easy to see that each summand in this decomposition corresponds to a different eigenvalue of [[GAMMA].sub.k]. Hence, we can use the Gamma operator to construct projection operators on each summand of the decomposition. So, the operator [P.sup.k.sub.i] (acting on [P.sub.k]) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

clearly satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 2.5.

In terms of spherical co-ordinates the Dunkl Dirac operator takes the form

[D.sub.k] = [[omega].bar]] ([[partial derivative].sub.r] + 1/r [[GAMMA].sub.k]).

Proof Again by means of Proposition 2.1, we obtain

r[[omega].bar]][D.sub.k] = -(E + [[GAMMA].sub.k]) = -(r[[partial derivative].sub.r] + [[GAMMA].sub.k]) or [D.sub.k] = [[omega].bar]] ([[partial derivative].sub.r] + 1/r [[GAMMA].sub.k]).

Lemma 2.3.

One has that [[GAMMA].sub.k][[[omega].bar]]] = ([mu] - l)[[omega].bar]], [[omega].bar]] [member of] [S.sup.m-1].

Proof. Taking into account Proposition 2.4 and Proposition 2.2 (iii), we find consecutively

[[GAMMA].sub.k][[[omega].bar]]] = [[GAMMA].sub.k][[[omega].bar]]/r] = 1/r [[GAMMA].sub.k][[x.bar]] = 1/r ([mu] - 1) [x.bar]] = ([mu] - 1) [[omega].bar]].

In the sequel the following positive weight function will play a crucial role:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is homogeneous of degree [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and invariant under reflections from the root system [R.sub.+].

A basic integral formula is the Stokes formula (see [3]), the proof of which heavily relies on the G-invariance of the weight function [w.sub.k]([x.bar]). Let [OMEGA] be a sufficiently smooth domain with boundary [GAMMA] = [partial derivative][OMEGA] and define the oriented surface element d[sigma]([x.bar]) on [GAMMA] by the Clifford differential form:

d[sigma]([x.bar]) = [m.summation over (j=1)] [(-1).sup.j] [e.sub.j][dx.sup.M\{j}],

where

[dx.sup.M\{j}] = d[x.sub.1] [conjunction] ... [conjunction] [d[x.sub.j]] [conjunction] ... [conjunction] d[x.sub.m], j = 1,2, ..., m.

If n([x.bar]) stands for the outward pointing unit normal at [x.bar] [member of] [GAMMA], then d[sigma]([x.bar]) = n([x.bar]) d[summation]([x.bar]), d[summation]([x.bar]) being the elementary Lebesgue surface measure.

Theorem 2.2 (Stokes and Cauchy theorem).

Let [OMEGA] be a sufficiently smooth domain invariant under the action of G, [GAMMA] = [partial derivative][OMEGA] and f, g [member of] [C.sup.[infinity]]([OMEGA]). Then

[[integral].sub.[OMEGA]][(f[D.sub.k]) g + f ([D.sub.k]g)] [w.sub.k]([x.bar])dV([x.bar]) = [[integral].sub.[GAMMA]] f [w.sub.k]([x.bar]) d[sigma]([x.bar])g.

Moreover, if f is right monogenic in [OMEGA] and g is left monogenic in [OMEGA], one has

[[integral].sub.[GAMMA]] f [w.sub.k]([x.bar])d[sigma]([x.bar]) g = 0.

We now have all the necessary results at our disposal in order to prove the orthogonality of the inner and outer Dunkl monogenics.

Theorem 2.3 (Orthogonality of Dunkl monogenics).

(i) Left inner Dunkl monogenics of different degree are orthogonal, i.e. for [M.sub.t] [member of] [M.sup.+.sub.l](t) and [M.sub.k] [member of] [M.sup.+.sub.l](k) with t [not equal to] k one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(ii) Left outer Dunkl monogenics of different degree are orthogonal, i.e. for [Q.sub.t] [member of] [M.sup.-.sub.l](t) and [Q.sub.k] [member of] [M.sup.-.sub.l] (k) with t [not equal to] k one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iii) Any left inner and left outer Dunkl monogenic are orthogonal, i.e. for all [Q.sub.t] [member of] [M.sup.-.sub.l](t) and [M.sub.k] [member of] [M.sup.+.sub.l](k) one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof.

(i)-(ii) This follows immediately from the fact that Dunkl harmonics of different degree are orthogonal, i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if k [not equal to] l (see [10],p. 177).

(iii) The proof is based on the Cauchy theorem with [OMEGA] the closed unit ball B(1) and hence [partial derivative][OMEGA] the unit sphere [S.sup.m-1].

Take [Q.sub.t] [member of] [M.sup.-.sub.l](t) and [M.sub.k] [member of] [M.sup.+.sub.l](k). As at each point [[omega].bar] [member of] [S.sup.m-1], n([[omega].bar]) = [[omega].bar], we have that d[sigma]([[omega].bar]) = [[omega].bar]dS([[omega].bar]) or dS([[omega].bar]) = -[[omega].bar] d[sigma]([[omega].bar]). Hence we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

As [Q.sub.t] [member of] [M.sup.-.sub.l](t), there exists [M.sub.t] [member of] [M.sup.+.sub.l](t) such that (see Proposition 2.3): [M.sub.t]([[omega].bar]) = [[omega].bar] [Q.sub.t]([[omega].bar]). Hence equation (4) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, as [bar.[M.sub.t]] is right monogenic in B(1), while [M.sub.k] is left monogenic in B(1), the Cauchy theorem yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3 Clifford-Gegenbauer polynomials on B(1) related to the Dunkl Dirac operator

3.1 The Clifford-Gegenbauer polynomials on B(1)

The Clifford-Gegenbauer polynomials on the unit ball B(1) were introduced by Cnops (see [4]). They are orthogonal on B(1) w.r.t. the weight function [(1 - [[absolute value of x].sup.2]).sup.[alpha]], [alpha] > -1.

Definition 3.1.

Let [P.sub.k] be a spherical monogenic of degree k, [alpha] [member of] R, [alpha] > -1 and t a positive integer. Then

[C.sup.m,[alpha].sub.t]([P.sub.k]) = [D.sub.[alpha]][D.sub.[alpha]+1] ... [D.sub.[alpha]+t-1][[P.sub.k]]

with

[D.sub.[alpha]] = (1 - [[absolute value of [x.bar]].sup.2])[[partial derivative].sub.[x.bar]] - 2([alpha] + 1)[x.bar]

is a Clifford-Gegenbauer polynomial. Here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] fixj is the so-called Dirac operator.

3.2 Definition Clifford-Gegenbauer polynomials on B(1) related to the Dunkl Dirac

By analogy with the previous subsection, we first introduce for [alpha] [member of] R and [alpha] > -1 the operator

[D.sub.[alpha]] = (1 - [[absolute value of [x.bar].sup.2])[D.sub.k] - 2([alpha] + 1)[x.bar].

The operator [D.sub.[alpha]] can be rewritten as

[D.sub.[alpha]] = [(1 - [[absolute value of [x.bar]].sup.2]).sup.-[alpha]] [D.sub.k][(1 - [[absolute value of [x.bar]].sup.2]).sup.[alpha] + 1]. (5)

Indeed, by means of the product rule (2), we obtain consecutively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We now define the Clifford-Gegenbauer polynomials on B(1) related to the Dunkl Dirac operator as follows:

Definition 3.2.

Let [M.sub.k] [member of] [M.sup.+.sub.l](k) and t a positive integer. Then

[C.sup.[alpha].sub.t,[mu]]([M.sub.k])([x.bar]) = [D.sub.[alpha]][D.sub.[alpha]+1][D.sub.[alpha]+2] ... [D.sub.[alpha] + t - 1][[M.sub.k]]

is a Clifford-Gegenbauer polynomial on B(1) of degree t associated with [M.sub.k].

Using Lemma 2.1 we see that the precise form of the polynomials [C.sup.[alpha].sub.t,[mu]]([M.sub.k]) depends only on the degree of the Dunkl-monogenic [M.sub.k], so we can write [C.sup.[alpha].sub.t,[mu]]([M.sub.k])([x.bar]) = [C.sup.[alpha].sub.t,[mu],k]([x.bar]([M.sub.k]).

The lower-degree polynomials take the following form:

[C.sup.[alpha].sub.0,[mu]]([M.sub.k]) = [M.sub.k]

[C.sup.[alpha].sub.1,[mu]]([M.sub.k]) = -2([alpha] + 1)[x.bar][M.sub.k]

[C.sup.[alpha].sub.2,[mu]]([M.sub.k]) = -2([alpha] + 2)(2[alpha] + 2 + 2k + [mu])[[x.bar].sup.2][M.sub.k] +([alpha] + 2)(2k + [mu])[M.sub.k].

It is clear that [C.sup.[alpha].sub.2s,[mu],k]([x.bar]) will only contain even powers of [x.bar], while [C.sup.[alpha].sub.2s + 1] k{x) will only contain odd powers of x.

3.3 Properties

Using the definition, we immediately obtain that the Clifford-Gegenbauer polynomials on B(1) satisfy the following recursion relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

The above result can be refined.

Proposition 3.1.

The Clifford-Gegenbauer polynomials satisfy the following recursion relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Proof. From (6) we immediately obtain (7), whereas

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next by means of the product rule (2) and Proposition 2.3, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, taking into account Proposition 2.5 and the fact that the angular Dunkl Dirac operator only acts on the angular co-ordinates, yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Furthermore, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we finally obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Taking into account (5), it is easily seen that there also exists a Rodrigues formula.

Theorem 3.1 (Rodrigues formula).

The Clifford-Gegenbauer polynomials on B(1) take the form

[C.sup.[alpha].sub.t,[mu]]([M.sub.k]) = [(1 - [[absolute value of [x.bar]].sup.2]).sup.-[alpha]] [D.sup.t.sub.k][[(1 - [[absolute value of [x.bar]].sup.2]).sup.[alpha]+t][M.sub.k]].

Moreover, the Clifford-Gegenbauer polynomials on B(1) satisfy an annihilation equation.

Theorem 3.2 (Annihilation equation). [C.sup.[alpha].sub.t,[mu]]([M.sub.k]) satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let us write down the expansion of the Clifford-Gegenbauer polynomials:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the recursion formula (6) and Lemma 2.1, we know that the following relations between the coefficients hold:

[[alpha].sup.2t,[alpha].sub.2i] = -(2i + 2k + [mu]) [a.sup.2t-1,[alpha]+1.sub.2i+1] - (2i + 2k + 2[alpha] + [mu]) [a.sup.2t-1,[alpha]+1.sub.2i-1] (9)

and

[a.sup.2t+1,[alpha].sub.2i+1] = -2([alpha] + 1 + i) [a.sup.2t,[alpha]+1.sub.2i] - (2i + 2) [a.sup.2t,[alpha]+1.sub.2i+2]. (10)

We need to prove

-2i [a.sup.2t,[alpha].sub.2i] = 2t(2[alpha] + 2t + [mu] + 2k) [a.sup.2t-1,[alpha]+1.sub.2i-1] (11)

and

-(2i + 2k + [mu]) [a.sup.2t+1,[alpha].sub.2i+1] = (2[alpha] + 2t + 2) (2t + [mu] + 2k) [a.sup.2t,[alpha]+1.sub.2i]. (12)

It is easy to check that the theorem holds for t = 0,1. Using (9) and (10) the theorem can then be proved by induction on t.

By acting on the above annihilation equation with [D.sub.[alpha]] we immediately obtain the differential equation satisfied by the Clifford-Gegenbauer polynomials.

Theorem 3.3 (Differential equation).

[C.sup.[alpha].sub.t,[mu]]([M.sub.k]) is a solution of the following differential equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The above equation should be compared with the classical differential equation of the Gegenbauer polynomials on the real line

(1 - [x.sup.2])[d.sup.2]/d[x.sup.2][C.sup.[lambda].sub.n](x) - (2[lambda] + 1) x d/dx [C.sup.[lambda].sub.n](x) + n(n + 2[lambda]) [C.sup.[lambda].sub.n](x) = 0

where it should be noticed that [lambda] = [alpha] + 1/2.

Moreover, combining the annihilation equation and the recursion formula (6) we obtain the following recurrence relation.

Theorem 3.4 (Recurrence relation).

[C.sup.[alpha].sub.t,[mu]]([M.sub.k]) satisfies the recurrence relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that this recursion formula is the Dunkl analogon of the classical one-dimensional Gegenbauer recurrence relation:

It is now possible to express the Clifford-Gegenbauer polynomials on B(1) in terms of the Jacobi polynomials on the real line.

Theorem 3.5 (Closed form).

The Clifford-Gegenbauer polynomials on B(1) can be written in terms of the Jacobi polynomials on the real line as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

with (a)p = a (a + 1) ... (a + p - 1) = [GAMMA](a + p)/[gamma](a) the Pochhammer symbol and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Recall that

[C.sup.[alpha].sub.2t,[mu]]([M.sub.k])([x.bar]) = [t.summation over (i=0)] [a.sup.2t,[alpha].sub.2i][[x.bar].sup.2i][M.sub.k].

By means of the annihilation equations (11) and (12) in terms of the coefficients [a.sup.s,[alpha].sub.i], we can write [a.sup.2t,[alpha].sub.2i] in terms of [a.sup.2t-2i,[alpha]+2i.sub.0] :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Now we look for an expression of [a.sup.2t,[alpha].sub.0]. By means of successively (9) and (12) we are able to write [a.sup.2t,[alpha].sub.0] in terms of [a.sup.0,[alpha]+2t.sub.0] = 1 :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Combining (15) and (16) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

from which we indeed obtain (13).

The formula for [a.sup.2t+1,[alpha].sub.2i+1] now follows from the annihilation equation (12): which leads to (14).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which leads to (14).

Let us mention the following corollary of the previous theorem.

Corollary 3.1.

The Clifford-Gegenbauer polynomials on B(1) satisfy

[C.sup.[alpha].sub.2t+1,[mu],k]([x.bar]) = -2([alpha] + 2t + 1) [x.bar] [C.sup.[alpha].sub.2t,[mu],k+1]([x.bar]).

By means of Theorem 3.5 we are able to prove the Dunkl-analogon of the classical one-dimensional three-term Gegenbauer recurrence relation:

n [C.sup.[lambda].sub.n](x) = 2(n + [lambda] - 1) x [C.sup.[lambda].sub.n-1](x) - (n + 2[lambda] - 2) [C.sup.[lambda].sub.n-2](x), n = 2,3, ...

Theorem 3.6 (Three-term recurrence relation).

[C.sup.[alpha].sub.t,[mu],k]([x.bar]) satisfies the three-term recurrence relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof This result follows from the following contiguous relations of the classical Jacobi polynomials on the real line (see [14]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3.4 Orthogonality

Let us consider the inner product:

[<f, g>.sub.[alpha]] = [[integral].sub.B(1)] [bar.f([x.bar])]g([x.bar])[(1 - [absolute value of [[x.bar].sup.2]]).sup.[alpha]][w.sub.k]([x.bar])dV([x.bar]).

Proposition 3.2.

The operators [D.sub.[alpha]] and [D.sub.k] are dual with respect to [<.,.>.sub.[alpha]], i.e. for all f, g [member of] [C.sub.1] (B(1))

Proof. By means of Theorem 2.2, we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

since the surface term clearly vanishes.

Using the above proposition, we are now able to prove an orthogonality relation for the Clifford-Gegenbauer polynomials on 6(1).

Theorem 3.7 (Orthogonality relation).

If s [not equal to] t or k [not equal to] l, then

[<[c.sup.[alpha].sub.t,[mu]]>([M.sub.k]), [C.sup.[alpha].sub.s,[mu]]([M.sub.l])>.sub.[alpha]] = 0.

Proof.

a) Suppose that s [not equal to] t and t > s (the case where t < s is similar). We obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

b) If s = t and k [not equal to] l, the result follows from the fact that the left inner Dunkl monogenics of different degree are orthogonal (see Theorem 2.3).

Using the orthonormality relation of the classical Jacobi polynomials, we are able to calculate the normalization constants.

Lemma 3.1.

We have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 3.1 (Gegenbauer polynomials on B(1) related to the Dunkl Laplacian).

Note that if we calculate [D.sub.[alpha]][D.sub.[alpha]+1] we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where we have used the product rule (2) and the relation (3).

As this operator is scalar, it makes sense to let it act on a Dunkl harmonic [H.sub.k] instead of a Dunkl monogenic. Hence, we can define

[C.sup.[mu],[alpha].sub.2t]([H.sub.k]) = [D.sub.[alpha]][D.sub.[alpha]+2][D.sub.[alpha]+4] ... [D.sub.[alpha]+2t-2][[H.sub.k]].

For these Gegenbauer polynomials associated with Hk we can also derive a Rodrigues formula, a differential equation, an annihilation equation and a recurrence relation. Moreover, in terms of the facobi polynomials on the real line, they take the following form

[C.sup.[mu],[alpha].sub.2t]([H.sub.k]) = [2.sup.2t] [([alpha] + t + 1).sub.t] t! [P.sub.([mu]/2+ k - 1,[alpha]).sup.t] (1 + 2[[x.bar].sup.2] [H.sub.k].

Note that special cases of these scalar polynomials and there associated weights have already been studied in [16,17].

4 Clifford-Gegenbauer polynomials on [R.sup.m] related to the Dunkl Dirac operator

4.1 The generalized Clifford-Gegenbauer polynomials

The generalized Clifford-Gegenbauer polynomials (see e.g. [8] and [2]) are defined as follows:

Definition 4.1.

Let [P.sub.k] be a spherical monogenic of degree k, [alpha] [member of] 1R and t a positive integer. Then

[G.sup.m,[alpha].sub.t]([P.sub.k]) = [D.sub.[alpha]][D.sub.[alpha]+1] ... [D.sub.[alpha]+t-1][[P.sub.k]]

with

[D.sub.[alpha]] = (1+ [absolute value of [[x.bar].sup.2]])[[partial derivative].sub.[x.bar]] + 2([alpha] + 11)[x.bar]

is a Clifford-Gegenbauer polynomial.

This second type of Clifford-Gegenbauer polynomials came into play while studying wavelets in the Clifford analysis setting. The polynomials, originally defined in a completely different way as by Cnops in [4], are the desired building blocks for new higher dimensional wavelet kernels (see [2]). They satisfy certain orthogonality relations on the whole of [R.sup.m] w.r.t. the weight function [(1 + [absolute value of [[x.bar].sup.2]]).sup.[alpha]], [alpha] [member of] R.

4.2 Definition Clifford-Gegenbauer polynomials on [R.sup.m] related to the Dunkl Dirac

By analogy with the previous subsection, we first introduce for [alpha] [member of] R the operator

[D.sub.[alpha]] = (1 + [absolute value of [[x.bar].sup.2]])[D.sub.k] + 2([alpha] + 11)[x.bar].

Similarly as in subsection 3.2 one can verify that the operator [D.sub.[alpha]] can be rewritten as

[D.sub.[alpha]] = [(1 + [absolute value of [[x.bar].sup.2]]).sup.-[alpha]][D.sub.k][(1 + [absolute value of [[x.bar].sup.2]]).sup.[alpha]+1]. (17)

We now define the Clifford-Gegenbauer polynomials on [R.sup.m] related to the Dunkl Dirac operator as follows:

Definition 4.2.

Let [M.sub.k] [member of] [M.sup.+.sub.l] (k) and t a positive integer. Then

[G.sup.[alpha].sub.t,[mu]]([M.sub.k])([x.bar]) = [D.sub.[alpha]][D.sub.[alpha]+1][D.sub.[alpha]+2] ... [D.sub.[alpha]+t-1][[M.sub.k]]

is a Clifford-Gegenbauer polynomial on [R.sup.m] of degree t associated with [M.sub.k].

Again using Lemma 2.1, it is clear that the precise form of the polynomials [G.sup.[alpha].sub.t,[mu]]([M.sub.k]) depends only on the degree of the Dunkl-monogenic [M.sub.k], so we can write [G.sup.[alpha].sub.t,[mu]]([M.sub.k])([M.sub.k])([x.bar]) = [G.sup.[alpha].sub.t,[mu],k]([x.bar]) ([M.sub.k]).

The lower-degree polynomials take the following form:

[G.sup.[alpha].sub.0,[mu]]([M.sub.k]) = [M.sub.k]

[G.sup.[alpha].sub.1,[mu]]([M.sub.k]) = 2([alpha] + 1)[x.bar][M.sub.k]

[G.sup.[alpha].sub.2,[mu]]([M.sub.k]) = 2([alpha] + 2)(2[alpha] + 2 + 2k + [mu])[[x.bar].sup.2][M.sub.k] - 2([alpha] + 2)(2k + [mu])[M.sub.k].

It is clear that [G.sup.[alpha].sub.2s,[mu],k]([x.bar]) will only contain even powers of [x.bar], while [G.sup.[alpha].sub.2s+1,[mu],k]([x.bar]) will only contain odd powers of [x.bar].

4.3 Properties

In this subsection we collect the properties of the Clifford-Gegenbauer polynomials on [R.sup.m]. As the proofs are similar as those in subsection 3.3, we omit them.

Using the definition, we immediately obtain that the Clifford-Gegenbauer polynomials on [R.sup.m] satisfy the following recursion relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

In particular, we obtain the following

Proposition 4.1.

The Clifford-Gegenbauer polynomials satisfy the following recursion relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

Taking into account (17), it is easily seen that there also exists a Rodrigues formula.

Theorem 4.1 (Rodrigues formula).

The Clifford-Gegenbauer polynomials on [R.sup.m] take the form

[G.sup.[alpha].sub.t,[mu]]([M.sub.k]) = [(1 + [absolute value of [[x.bar].sup.2]]).sup.-[alpha]] [D.sup.t.sub.k][[(1 + [absolute value of [[x.bar].sup.2]]).sup.[alpha]+t] [M.sub.k]].

Moreover, the Clifford-Gegenbauer polynomials on M.m satisfy an annihilation equation.

Theorem 4.2 (Annihilation equation). [G.sup.[alpha].sub.t,[mu]]([M.sub.k]) satisfies

[D.sub.k][[G.sup.[alpha].sub.t,[mu]]([M.sub.k])] = -C([alpha], t, [mu], k)[G.sup.[alpha]+1.sub.t-1,[mu]]([M.sub.k]).

From the above annihilation equation we obtain the differential equation

Theorem 4.3 (Differential equation).

[G.sup.[alpha].sub.t,[mu]]([M.sub.k]) is a solution of the following differential equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, combining the annihilation equation and the recursion formula (18) we find the following recurrence relation.

Theorem 4.4 (Recurrence relation).

[G.sup.[alpha].sub.t,[mu]]([M.sub.k]) satisfies the recurrence relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can now express the Clifford-Gegenbauer polynomials on [R.sup.m] in terms of the Jacobi polynomials on the real line.

Theorem 4.5 (Closed form).

The Clifford-Gegenbauer polynomials on [R.sup.m] can be written in terms of the Jacobi polynomials on the real line as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, the previous theorem enables us to prove the following result.

Theorem 4.6 (Three-term recurrence relation).

[G.sup.[alpha].sub.t,[mu],k]([x.bar]) satisfies the three-term recurrence relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4.4 Orthogonality

The orthogonality of the Clifford-Gegenbauer polynomials on [R.sup.m] must be treated in a completely different way than the orthogonality of their counterparts on the unit ball B(1). It must be expressed in terms of a bilinear form instead of an integral (as was also done in [5,8] for the super resp. classical case).

Let us start by computing the following integral on the Euclidean space [R.sup.m] with [M.sub.k] and [M.sub.l] left solid inner Dunkl monogenics of order k, respectively l:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

using the Beta function

B(x,y) = [[integral].sup.+[infinity].sub.0] [u.sup.x-1] [(1 + u).sup.-x-y] du, Re(x) > 0 , Re(y) > 0.

Naturally, the last equality in (21) only holds if 2[alpha] < - (k + s + t + l + [mu]). The above integral consists of two parts: a radial part and an angular part which is an integration over the unit sphere. If we consider e.g. the case s = 2a, t = 2b, the angular integral simplifies to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The case where s and t are both odd, i.e. s = 2a + 1 and t = 2b + 1, yields the same result as above.

On the other hand, if s and t have different parity, e.g. s = 2a and t = 2b + 1, the integral over the unit sphere vanishes, since outer and inner Dunkl monogenics are orthogonal (see Theorem 2.3).

In what follows, we will restrict ourselves to spaces of polynomials of the type

R([M.sub.k]) = {[p.sub.n]([x.bar]) [M.sub.k]([x.bar]) = [n.summation over (j=0)] [a.sub.j] [[x.bar].sup.j] [M.sub.k]([x.bar]) | n [member of] N, [a.sub.j] [member of] R}

where [M.sub.k] is a left solid inner Dunkl monogenic of degree k, fixed once and for all, which satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Inspired by the previous calculations and using the analytic continuation of the Gamma function, we are led to the following definition of a bilinear form on R([M.sub.k]).

Definition 4.3.

The bilinear form [<.,.>.sub.[alpha]] (parameterized by [alpha]) on R([M.sub.k]) is defined by linear extension of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that this bilinear form is symmetric, but not always positive definite. Moreover, this bilinear form is well-defined if and only if [alpha] [member of] N and [mu]/2 + [alpha] [not member of] [+ or -] N, due to the singularities z = -n, n [member of] N of the Gamma function [GAMMA](z). Note that due to Lemma 2.1 we have that [D.sub.[alpha]][R([M.sub.k])] [subset] R([M.sub.k]).

Proposition 4.2.

The operators [D.sub.k] and [D.sub.[alpha]] are dual with respect to [<.,.>.sub.[alpha]], i.e.

[<[D.sub.[alpha]][[p.sub.i] [M.sub.k]], [p.sub.j] [M.sub.k]>.sub.[alpha]] = [<[p.sub.i] [M.sub.k], [D.sub.k][[p.sub.j] [M.sub.k]]).sub.[alpha]+1]

with [p.sub.i] [M.sub.k], [p.sub.j] [M.sub.k] [member of] R([M.sub.k]).

Proof. It is sufficient to prove the result for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By means of the definitions of the operator [D.sub.[alpha]] and the bilinear form [<.,.>.sub.[alpha]], we have consecutively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As [GAMMA](z + 1) = z [GAMMA](z), we can further simplify the above result

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The other three cases are treated similarly.

Now we come to the orthogonality relation of the Clifford-Gegenbauer polynomials.

Theorem 4.7.

If s [not equal to] t, then

[<[G.sup.[alpha].sub.s,[mu]]([M.sub.k]), [G.sup.[alpha].sub.t,[mu]]([M.sub.k])>.sub.[alpha]] = 0.

Proof. Suppose that s > t. The case where s < t is similar. By means of the above proposition and Lemma 2.1 we obtain consecutively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

References

[1] F. Brackx, R. Delanghe and F. Sommen, Clifford analysis. Pitman Publishers, Boston-London-Melbourne, 1982.

[2] F. Brackx, N. De Schepper and F. Sommen, The Clifford-Gegenbauer Polynomials and the Associated Continuous Wavelet Transform. Integral Transform. Spec. Funct. 15(5) (2004), 387-404.

[3] P. Cerejeiras, U. Kahler, G. Ren, Clifford analysis for finite reflection groups. Complex Variables and Elliptic Equations, 51:5 (2006), 487-495.

[4] J. Cnops, Orthogonal functions associated with the Dirac operator. PhD thesis, Ghent University, 1989.

[5] H. De Bie and F. Sommen, Hermite and Gegenbauer polynomials in superspace using Clifford analysis. J. Phys. A: Math. Theor. 40 (2007), 10441-10456.

[6] H. De Bie, An alternative definition of the Hermite polynomials related to the Dunkl laplacian. SIGMA 4 (2008), 093,11 pages, arXiv:0812.4819.

[7] R. Delanghe, F. Sommen and V. Soucek, Clifford algebra and spinor-valued functions. Kluwer Academic Publishers, Dordrecht, 1992.

[8] N. De Schepper, The Generalized Clifford-Gegenbauer Polynomials Revisited. Adv. appl. Clifford Alg. 19(2) (2009), 253-268.

[9] C. F. Dunkl, Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311,1 (1989), 167-183.

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

[11] G.J. Heckman, A remark on the Dunkl differential-difference operators. Barker, W., Sally, P. (eds.) Harmonic analysis on reductive groups. Progress in Math. 101, pp. 181 -191. Basel: Birkhauser Verlag 1991.

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

[13] M. A. Olshanetsky and A. M. Perelomov, Quantum integrable systems related to Lie algebras. Phys. Rep. 94 (1983), 313-404.

[14] W. Magnus, F. Oberhettinger and R.-P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag: Berlin-HeidelbergNew York; 1966.

[15] B. Orsted, P. Somberg and V. Soucek, The Howe duality for the Dunkl version of the Dirac operator. Adv. appl. Clifford alg. 19 (2009), 403-415.

[16] Y. Xu, Summability of Fourier orthogonal series for Jacobi weight on a ball in [R.sup.d]. Trans. Amer. Math. Soc. 351 (1999), 2439-2458.

[17] Y. Xu, Orthogonal polynomials on the ball and on the simplex for weight functions with reflection symmetries. Const. Approx. 17 (2001), 383-412.

H. De Bie *

N. De Schepper

* Postdoctoral Fellow of the Research Foundation--Flanders (FWO)

Received by the editors December 2009.

Communicated by F. Brackx.

2000 Mathematics Subject Classification : 30G35 (primary), 33C80,33C45 (secondary).

Clifford Research Group--Department of Mathematical Analysis

Faculty of Engineering--Ghent University

Krijgslaan 281,9000 Gent, Belgium

E-mail:Hendrik.DeBie@UGent.be, nds@cage.ugent.be
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Date:May 1, 2011
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