RODRIGUES TYPE FORMULA FOR ORTHOGONAL POLYNOMIALS ON THE UNIT BALL

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1 RODRIGUES TYPE FORMULA FOR ORTHOGONAL POLYNOMIALS ON THE UNIT BALL YUAN XU Abstract. For a class of weight function invariant uner reflection groups on the unit ball, a family of orthogonal polynomials is efine via a Rorigues type formula using the Dunkl operators. Their properties an their relation with several other bases are explore.. Introuction Let Π = Rx,..., x ] enote the space of polynomials in variables an Π n enote the subspace of polynomials of egree at most n. We will use the stanar multiinex notation. For α N 0 an x R, x α = x α xα is a monomial of egree α = α α. The classical orthogonal polynomials on the unit ball B = {x : x }, where x is the Eucliean norm of x, are orthogonal with respect to the weight function.) W µ x) = x 2 ) µ /2, x B, µ > /2. The stuy of these polynomials can be trace back to Hermite, see ] an 5, Chapt. 2]. Among the first orthogonal bases on B being stuie is the one given by the Rorigues type formula.2) U α x) = x 2 ) µ+/2 α x 2 ) n+µ /2, α N 0, where i = / x i an α = α α, an a family of polynomials that are biorthogonal to U α. The purpose of this paper is to consier the analogues of U α for orthogonal polynomials on B with respect to a class of weight functions invariant uner a reflection group. We first efine the weight function. Let G be a finite reflection group with a fixe positive root system R +. Let σ v enote the reflection along v R +, that is, xσ v = x 2 x, v / v 2 for x R with the inner prouct x, y = i= x iy i. Thus, G is a finite orthogonal group generate by the reflections {σ v : v R + }. Let κ be a multiplicity function efine on R +, that is, κ : R + R is a G-invariant function. We assume that κv) 0 for all v R +. Then the function.3) h κ x) = x, v κv), x R, is a positive homogeneous G-invariant function of orer γ := γ κ = κ v. For example, if G is the symmetric group, then h κ x) = i<j x i x j κ an Date: February. 3, Mathematics Subject Classification. 33C50, 42C0. Key wors an phrases. orthogonal polynomials on a unit ball, Rorigues type formula, biorthogonal polynomials, reflection invariant weight function. Work partially supporte by the National Science Founation uner Grant DMS

2 2 YUAN XU γ = )κ/2. In this paper we consier the reflection invariant weight function W κ,µ on the unit ball.4) W κ,µ x) = h 2 κx) x 2 ) µ /2, x B, µ > /2 where h 2 κ is as in.3). If κ = 0 then W κ,µ x) becomes W µ x) in.). The weight function h κ was introuce by Dunkl for the purpose of stuying the orthogonal structure for polynomials on the unit sphere S = {x R : x = } with respect to the measure h 2 κx)ω, where ω is the rotation invariant measure on S. The main ingreient in the stuy is a family of commuting ifferentialifference operators, D i, calle Dunkl s operators 2]), D i fx) = i fx) + fx) fxσ v ) κ v e i, v, i, x, v where e i = 0,..., 0,, 0,..., 0) with in the i-th component. The set {D i : i } generates a commutative algebra of operators containing the h-laplacian h = D D 2. Let P n enote the space of homogeneous polynomials of egree n in variables. If p Pn then px)qx)h 2 x)ω = 0, q Π n, S if an only if h p = 0. The space Hnh 2 κ) := Pn ker h is calle the space of h- harmonic polynomials of egree n. If κv) = 0, it agrees with the space of orinary harmonic polynomials Hn := Pn ker, where enote the orinary Laplacian = For the extensive stuy of h-harmonics an the relate work, see 4] an the references therein. The theory of h-harmonics has also mae it possible to stuy the orthogonal structure of polynomials with respect to W κ,µ on B. Initial work in this irection has been carrie out in 2, 3], which has lea to new results even in the case of the classical weight function W µ. In the present paper we show that the orthogonal polynomials U α x) in.2) efine by the Rorigues type formula can be extene to the reflection invariant weight function W κ,µ. That is, we efine U α as follows: Definition.. For α N 0, efine U α x) = x 2 ) µ+/2 D α x 2 ) α +µ /2. We will show that U α x) so efine are orthogonal polynomials with respect to W κ,µ on B. Properties of these polynomials an their relations with other orthogonal bases will be iscusse. Let us mention that recently Dunkl an Dunkl-Cherenik operators have been use to give the Rorigues type formula for nonsymmetric Hermite an Laguerre polynomials with respect to reflection invariant weight functions 8, 0]) an for Maconal s polynomials 7]). 2. Rorigues type formula an Orthogonal bases For n N 0 let V nw κ,µ ) enote the subspace of polynomials of egree exactly n in Π that are orthogonal with respect to W κ,µ. A polynomial p V nw κ,µ ) if p Π n an B px)qx)w κ,µ x)x = 0 for all q Π n. We show U α V nw κ,µ ). The efinition of U α in Definition. shows that it epens on κ an µ. In the following proposition, we write U µ α := U α to emphasis its epenence on µ.

3 ORTHOGONAL POLYNOMIALS ON THE UNIT BALL 3 Proposition 2.. Let α N 0 an α = n. Then U µ α is a polynomial of egree n an we have a recursive formula U µ α+e i x) = 2µ + )x i U α µ+ x) + x 2 )D i U α µ+ x). Proof. Since x 2 ) a is invariant uner the reflection group, a simple computation shows that D i U α µ+ x) = i x 2 ) µ /2] D α x 2 ) α +µ+/2 + x 2 ) µ /2 D α+ei x 2 ) α +µ+/2 =2µ + )x i x 2 ) U α µ+ x) + x 2 ) U µ α+e i x), which proves the recursive relation. That U α µ is a polynomial of egree α is a consequence of this relation use inuction if necessary). The following proposition plays a key role in proving that U α is in fact an orthogonal polynomial with respect to W κ,µ on B. Proposition 2.2. Assume that p an q are continuous function an p vanishes on the bounary of B. Then D i px)qx)h 2 κx)x = px)d i qx)h 2 κx)x. B B Proof. The proof is similar to the the proof of Lemma 3.7 of 3] Theorem 5..8 of 4]). We shall be brief. Assume κ v. Analytic continuation can be use to exten the range of valiity to κ v 0. Integration by parts shows i px)qx)h 2 κx)x = px) i qx)h 2 κx))x B B = px) i qx)h 2 κx)x 2 px)qx)h κ x) i h κ x)x. B B Let D i enote the ifference part of D i, D i = i + D i. For a fixe root v, a simple computation shows that D i px)qx)h 2 κx)x = px) qx) + qxσ v ) κ i v i h 2 B B x, v κx)x. Aing these two equations an using the fact that h κ x) i h κ x) = κ v v i v, x h2 κx) completes the proof. Theorem 2.3. For α N 0, the polynomials U α are elements in V α W κ,µ). Proof. Let n = α. For each β N 0 an β < n, we claim that 2.) D β x 2 ) n+µ /2 = x 2 ) n β +µ /2 Q β x)

4 4 YUAN XU for some Q β Π n. This follows from inuction. The case β = 0 is trivial with Q 0 x) =. Suppose the equation is true for β < n. Then ] D β+ei x 2 ) n+µ /2 = D i x 2 ) n β +µ /2 Q β x) = i x 2 ) n β +µ /2] Q β x) + x 2 ) n β +µ /2 D i Q β x) = x 2 ) n β +µ 3/2 2n β + µ /2)x i Q β x) + x 2 )D i Q β x) ] = x 2 ) n β+ei +µ /2 Q β+ei x), where Q β+ei, as efine above, is clearly a polynomial of egree β +, which completes the inuctive proof. This formula shows, in particular, that D β x 2 ) n+µ /2 is a function that vanishes on the bounary of B if β < n. For any polynomial p Π n, using the Proposition 2.2 repeately gives U α x)px)h 2 κx) x 2 ) µ /2 x = D α x 2 ) n+µ /2] px)h 2 κx)x B B = ) n B D α px) x 2 ) n+µ /2 h 2 κx)x = 0, since D i : P n P n, which implies D α px) = 0 as p Π n. By efinition, U α is orthogonal to polynomials of lower egree. However, if α = β, U α an U β are not necessarily orthogonal. In fact, they are not an there is another family of orthogonal polynomials in VnW κ,µ ) that are biorthogonal to U α. For its efinition, we nee the projection operator proj B n : Pn VnW κ,µ ). This operator has an explicit formula 2]). For p Pn, 2.2) proj B px) = 4 j j h j! n λ µ + ) px), j 0 j n/2 where λ = γ κ + )/2 an a) j = aa+)... a+j ) is the Pochhammer symbol. Let V κ be the intertwining operator between the family of partial erivatives an Dunkl s operators, which is a linear operator uniquely etermine by 3]) V κ =, V κ P n = P n, D i V κ = V κ i, i. The explicit formula of V κ, however, is known only in the case of the group Z 2 an S 3, the symmetric group of three variables. For each β N 0, the function V κ x β is a homogeneous polynomial of egree β. Define R β x) := proj B n V κ {x β } = V κ 4 j j x β, j! n λ µ + ) j 0 j n/2 where the secon equality follows from 2.2) an the fact that h V κ = V κ. Clearly R β is an orthogonal polynomial of egree n with respect to W κ,µ an {R β : β = n} is a basis of V nw κ,µ ). It turns out that R β an U α are biorthogonal. For α N 0, write α! = α!... α!. Theorem 2.4. The two families {U α : α = n, α N 0} an {R α : α = n, α N 0} are biorthogonal with respect to W κ,µ on B ; more precisely, a κ U α x)r β x)w κ,µ x)x = A α δ α,β, A α = ) B n µ + /2) n α!. λ + µ + /2) n

5 ORTHOGONAL POLYNOMIALS ON THE UNIT BALL 5 In particular, {U α : α = n} is a basis for V nw κ,µ ). Proof. Since both U α an V β are orthogonal polynomials with respect to W κ,µ, we only nee to consier the case that they are both polynomials of egree n. Let α, β N 0 an α = β. Following the proof of Proposition 2.3, we have a κ B U α x)r β x)h 2 κx) x 2 ) µ /2 x = ) n a κ B D α R β x)] x 2 ) n+µ /2 h 2 κx) = ) n a κ B D α V κ x β] x 2 ) n+µ /2 h 2 κx) = δ α,β α! ) n a κ B x 2 ) n+µ /2 h 2 κx), since R β x) = x β + lower egree polynomials an D α V κ x β = V κ α x β = α!δ α,β V κ = α!δ α,β. The constant is compute by x 2 ) n+µ /2 h 2 κx) = r +2γ r 2 ) µ /2 r h 2 κx )ωx ) B 0 S Γγ + /2)Γµ + /2) = h 2 2Γγ + µ + + )/2) κx )ωx ) S = µ + /2) n λ + µ + /2) n a κ, since a κ is just the above integral with n = 0. Finally, it follows from the biorthogonality that the set {U α : α = n} is linearly inepenent, so that it is a basis of VnW κ,µ ). For our next property of U α x), we will nee the reproucing kernel P n x, y) of V nw κ,µ ). This kernel is characterize by the property that a κ B P x, y)qy)w κ,µ y)y = qx), q V nw κ,µ ). It is shown in 2] that the kernel enjoys a close formula given by 2.3) P n x, y) = n + λ + µ λ + µ c µ V κ Cn λ+µ x, + t x 2 y 2 ) t 2 ) µ t ] y), where c µ = / t2 ) µ t an V κ acts on the variable represente by the ot in the formula. This close formula for the reproucing kernel has been use to stuy summability of the Fourier orthogonal expansion on the unit ball. It follows from the biorthogonal property of {U α } an {R β } that 2.4) P n x, y) = A α R α x)u α y). α =n Inee, multiplying the right han sie of the above equation by R β y) an integrating, the result is R β y) by the biorthogonality, which shows that the right han sie satisfies the reproucing property. Since the kernel is unique, it has to equal the left han sie.

6 6 YUAN XU Let K n x, y) = V κ x, y n )/n!. It is known 3]) that 2.5) K n x, D y) )py) = px), p P n, where D y) means that D i is taken with respect to the variable y an K n x, D y) ) is the operator forme by replacing y i by D y) i in K n x, y). A bilinear form p, q h is efine on Pn by p, q h = pd)qx), p, q P n. This form is symmetric in the sense that p, q h = q, p h 3]). We have the following lemma that will be useful below. Lemma 2.5. Let α N 0 with α = n an q P j with j < n. Then D y) ) α qy)v y) κ x, y n j] = n j)!qd)x α. Proof. Let β N 0 an β = j. As a function in y, y β V κ x, y n j is a polynomial in P n. Using the fact that the bilinear form is symmetric, we get D y) ) α y β V κ x, y n j] /n j)! = y α, y β K n β x, y) h = y β K n β x, y), y α h = K n β x, D y) )D β y α = D β x α, where the last step follows from the equation 2.5) with py) = D β y α an n replace by n j. Proposition 2.6. Let α N 0 an α = n. Then U α x) = 0 2j n 2 n µ + /2) n 2 2j µ + /2) j j! )n j x 2 ) j j h xα. In particular, U α x) = ) n 2 n µ + /2) n x α + x 2 )Qx), where Q Π n 2. Proof. Since D α R β y) = α!δ α,β, applying D y) ) α to the equation 2.4) gives α!a α U α x) = D y) ) α P n x, y). We then use the expression of P n x, y) in 2.3). Since the leaing coefficient of Cnt) λ is λ) n 2 n /n!, it follows that P n x, y), as a function of y, satisfies P n x, y) = n + λ + µ λ + µ) n 2 n c µ λ + µ n! V κ x, + t ] x 2 y 2 ) n t 2 ) µ t y) +... = λ + µ + ) n2 n n! c µ 0 2j n + polynomial of lower egree in y, ) n b j x 2 ) j V κ y 2 ) j x, n 2j] y) 2j where we have use the binomial formula an the fact that tj t 2 ) µ t = 0 if j is o, an b j is given by b j = c µ t 2j t 2 ) µ t = /2) j. µ + /2) j

7 ORTHOGONAL POLYNOMIALS ON THE UNIT BALL 7 Consequently, since D α px) = 0 for any p Π m, m < α, we have α!a α U α x) = λ + µ + ) n2 n ) n /2)j c µ x 2 ) j n! 2j µ + /2) j 0 j n Using Lemma 2.5 with qy) = y 2j, we have D y) ) α y 2 ) j V κ x, n 2j ) y) ]. D y) ) α y 2 ) j V κ x, n 2j ) y) ] = ) j D y) ) α y 2j V κ x, n 2j ) y) ] Putting these equations together we conclue U α x) = A α λ + µ + ) n 2 n α!n! c µ 0 j n = ) j n 2j)! j h xα. ) j n!/2) j x 2 ) j j h 2j)!µ + /2) xα. j Using the ientity 2j)! = 2 2j /2) j j! to simplify the constants completes the proof. We note that the expansion is still implicit in a way, since the computation of j h xα can be ifficult. In the case of classical orthogonal polynomials on B, h becomes, the multinomial formula gives j x α = ) 2 j x α = β =j j! β! 2β x α = β =j j! α! β! α 2β)! xα 2β, from which the explicit formula of U α x) for W µ follows. Since Dunkl s operators commute, the multinomial formula can also be use to expan h x α. However, we o not have a nice formula for D β x α. This is a polynomial of egree α β in x. For α = β it is equal to x α, x β h an is easily seen to be a polynomial of κ. However, we o not know an explicit formula for this quantity. We mention the following recursive formula, which can be use to compute D β x α of lower egree. Proposition 2.7. Let α, β N 0, β α. If α i > 0, then D β x α = x i D β x α ei + β D β ei x α ei + κ v v, e i P β,v D)x α ei, where P β,v y) = y β yσ v ) β )/ y, σ v is a homogeneous polynomial of egree β. Proof. We use the prouct rule of Dunkl s operators 4, p. 57]) D i fg)x) = gx)d i fx) + fx) i gx) + κ v fxσ v ) v, e i gx) gxσ v), x, v Using the fact that D i K n x, y) = x i K n x, y), the prouct rule with gy) = y β an fy) = K α β x, y) shows y β K α β x, y) ] =y β x i K α β x, y) + K α β x, y)β i y β ei D y) i + κ v K α β x, yσ v ) v, e i yβ yσ v )β. x, v

8 8 YUAN XU Let φ α,β x) = D y) i ) α y β K α β x, y) ]. Since K n xw, yw) = K n x, y) for all w G, applying D y) i ) α ei to the above equation gives φ α,β x) = x i φ α ei,βx) + β i φ α ei,β e i x) + κ v v, e i c τ φ α ei,τ xσ v ), τ = β where y β yσ v ) β )/ x, v = τ = β c τ y τ. By Lemma 2.5, φ α,β x) = D β x α, which gives the state formula. 3. Relation to an orthonormal basis The biorthogonality of two bases is useful for fining the orthogonal expansions of a function. For example, for f L 2 W κ,µ ), the Fourier orthogonal expansion of f in terms of the basis {U α } is given by fx) = n=0 α =n b α,n A α U α x), b α,n = a κ B fx)r α x)w κ,µ x)x, where the equality hols in L 2 norm. That the coefficient b α,n is given as above can be easily seen upon integrating the expansion of fx)r α x) over B. Thus, the biorthogonality provies an easy way to compute the Fourier coefficient b α,n in the expansion with respect to U α x). Note that R α epens on the intertwining operator V κ. The following formula prove in ] is useful for computing integrals of V κ f, which works espite the lack of explicit formula for V κ. Lemma 3.. For a continuous function f on B, S V κ fx)h 2 κx)ωx) = B κ B fx) x 2 ) γ x where the constant B κ can be etermine by setting fx) =. We use this lemma to work out the expansion in one particular case. Let C n µ,τ) t) be polynomials efine by the generating function c µ 2xt + t 2 ) µ + t) t2 ) τ t = n=0 C µ,τ) n x)t n. These are orthogonal polynomials with respect to the weight function w µ,τ t) = t 2τ t 2 ) µ /2 on, ] an they are relate to the Jacobi polynomials P n α,β) t). In particular, 3.) C µ,τ) 2n t) = µ + τ) n τ + /2) n P µ /2,τ /2) n 2t 2 ). It is known that an orthonormal basis of VnW κ,µ ) is given by { } 3.2) f j,β x) := m Cµ,n 2j+λ) j 2j x )Sβx) h : Sβ h Hn 2jh 2 κ), 0 2j n, in which c 2 j is the L 2 w λ,µ,, ]) norm of C µ,n 2j+λ) 2j t) an {Sβ h } is an orthonormal basis of Hn 2j h2 κ). In particular, the polynomial 2n x ) is an element of V2nW κ,µ ); hence it can be written in terms of U α x).

9 ORTHOGONAL POLYNOMIALS ON THE UNIT BALL 9 Proposition 3.2. The following expansion hols, 3.3) 2n x ) = 2n ) n! µ + /2) 2n 2 2n β =n β! U 2βx). Proof. The fact that {U α : α = 2n} is a basis of V 2nW κ,µ ) shows that we can write 2n x ) = α =2n b α A α U α x). Using the biorthogonality in Proposition 2.4 an the fact that R κ x) = V κ x α +..., we can compute the coefficient b α as follows: b α = a κ,µ x )R α x)h 2 κx) x 2 ) µ x = a κ,µ = b κ,µ 2n B 2n B 0 x )V κ x α ] h 2 κx) x 2 ) µ x r 2n+ +2γ 2n r) r 2 ) µ /2 r c h S V κ { } α )x )h 2 κx )ωx ), using the polar coorinates x = rx, r > 0 an x S, where b κ,µ = / 0 r2λ r 2 ) µ /2 r an c h = / h 2 S κx )ωx ). The first integral can be compute using the L 2 norm an the leaing coefficient k n µ,τ) of C n µ,τ) t) cf. 4, p. 27]), b κ,µ 0 r 2n+ +2γ 2n r) r 2 ) µ /2 r = b κ,µ k µ,τ) ) 0 2n ] 2 r) r 2 ) µ /2 r = µ + /2) nλ + µ) n, λ + µ + ) 2n The integral of V κ x α can be compute using Lemma 3., c h V κ { } S α )x )h 2 κx )ω = a κ x α x 2 ) γ x, B where a κ = / x 2 ) γ x, which shows that the integral is zero if one B of α i is o. If α = 2β, then the integral over B can be evaluate using polar coorinates an the result is c h V κ { } α )x )h 2 κx )ω = /2) β. S λ + /2) β Putting these equations together, we have prove that 2n x ) = µ + /2) n µ + λ) n /2) β A 2β µ + λ + ) 2n λ + /2) U 2βx) β β =n Using the formula of 2n ) 4, p. 27]) an simplifying the coefficients completes the proof.

10 0 YUAN XU Let us write the expansion explicitly. Using the multinomial formula an the Rorigues type formula of U β x), we see that the equation 3.3) gives 2n x ) = 2n ) = µ + /2) 2n 2 2n x 2 ) µ+/2 β =n n! β! D2β x 2 ) 2n+µ /2 µ + /2) 2n 2 2n x 2 ) µ+/2 n h x 2 ) 2n+µ /2. In particular, if κ = 0 an =, so that λ = 0, then C µ,0) 2n x ) becomes the Gegenbauer polynomial C µ 2n x) an the above formula is precisely the Rorigues formula for the Gegenbauer polynomials. Furthermore, in polar coorinates, the h-laplacian can be written as 3.4) h = 2 r 2 + 2λ + r r + r 2 h,0, where h,0 is the spherical part of the operator which has h-harmonics as eigenfunctions. More precisely, if S h n H nh 2 κ) then 3.5) h,0 S h nx) = nn + 2λ)S h nx). This leas to the following Rorigues type formula for 2n t). Proposition 3.3. For n 0, 2n r) 2n ) = µ + /2) 2n 2 2n r2 ) µ+/2 2 r 2 + 2λ r ) n r 2 ) 2n+µ /2. r By the relation 3.), this gives a Rorigues type formula for the Jacobi polynomials, which can be rewritten as P n α,β) 2r 2 α + ) n ) = α + ) 2n n!2 2n r2 ) α 2 r 2 + 2β + ) n r 2 ) 2n+α. r r In fact, this formula was iscovere by Koornwiner 6] in the course of giving an analytic proof for his aition formula for the Jacobi polynomials. It is possible to erive a Rorigues type formula for other elements in the orthonormal basis 3.2). We have the following, Proposition 3.4. Let S h n 2j be an h-spherical harmonic in H nh 2 κ). Then C µ,n 2j+λ) 2j x ) C µ,n 2j+λ) 2j ) Sh n 2jx) = µ + /2) 2j 2 2j x 2 ) µ+/2 j h ] x 2 ) 2j+µ /2 Sn 2jx) h. Proof. Let g be a ifferentiable function efine on 0, ] an Sm h Hmh 2 κ). We claim that h g x )S h m x) ] ) = Smx) h 2 2λ + 2m + + gr), r = x. r2 r r

11 ORTHOGONAL POLYNOMIALS ON THE UNIT BALL Inee, since Sm h is a homogeneous polynomial of egree r, Smx) h = r m Smx h ) uner the polar coorinates. Therefore, using 3.4) an 3.5), h g x )S h m x) ] 2 = r 2 + 2λ + ) r m gr)) S h r r mx ) mm + 2λ)gr)r m 2 Smx h ) ) ] 2 2λ + 2m + = + gr) r m S h r2 r r mx ), where the secon equality follows from a simple computation. Clearly, we can use the formula repeately to get j h g x )S h m x) ] ) = Smx) h 2 j 2λ + 2m + + gr). r2 r r Applying the above formula with gr) = r 2 ) 2j+µ /2 an m = n 2j, the state formula follows from Proposition 3.3. Acknowlegement. The author thanks a referee for his/her careful review. References ] P. Appell an J. K. e Fériet, Fonctions hypergéométriques et hypersphériques, Polynomes Hermite, Gauthier-Villars, Paris, ] C. Dunkl, Differential-ifference operators associate to reflection groups, Trans. Amer. Math. Soc ), ] C. Dunkl, Integral kernels with reflection group invariance, Cana. J. Math ), ] C. F. Dunkl an Yuan Xu, Orthogonal polynomials of several variables, Cambrige Univ. Press, ] A. Erélyi, W. Magnus, F. Oberhettinger an F. G. Tricomi, Higher transcenental functions, McGraw-Hill, Vol 2, New York, ] T. Koornwiner, Jacobi polynomials, III. An analytic proof of the aition formula, SIAM J. Math. Anal., 6 975), p ] L. Lapointe an L. Vinet, Rorigues formulas for the Maconal polynomials, Av. Math ), ] A. Nishino, H. Ujino an M. Waati, Rorigues formula for the nonsymmetric multivariable Laguerre polynomial, J. Phys. Soc. Japan ), ] G. Szegő, Orthogonal polynomials, 4th e., American Mathematical Society Colloquium Publication 23, American Mathematical Society, Provience, RI, ] H. Ujino an M. Waati, Rorigues formula for the nonsymmetric multivariable Hermite polynomial, J. Phys. Soc. Japan ), ] Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associate to reflection groups, Proc. Amer. Math. Soc ), ] Yuan Xu, Orthogonal polynomials on the ball an on the simplex for weight functions with reflection symmetries, Constr. Approx., 7 200), ] Yuan Xu, Orthogonal polynomials an summability in Fourier orthogonal series on spheres an on balls, Math. Proc. Cambrige Phil. Soc ), Department of Mathematics, University of Oregon, Eugene, Oregon aress: yuan@math.uoregon.eu