A formula for the nonsymmetric Opdam's hypergeometric function of type A_2
aa r X i v : . [ m a t h . C A ] A p r A formula for the nonsymmetric Opdam’shypergeometric function of type A Béchir Amri and Mounir Bedhiafi
Université Tunis El Manar, Faculté des sciences de Tunis,Laboratoire d’Analyse Mathématique et Applications,LR11ES11, 2092 El Manar I, Tunisie. e-mail: [email protected], bedhiafi[email protected]
Abstract
The aim of this paper is to give an explicit formula for the nonsymmetricHeckman-Opdam’s hypergeometric function of type A . This is obtained bydifferentiating the corresponding symmetric hypergeometric function. Keywords . Root systems, Cherednik operators, Hypergeometric functions.
Mathematics Subject Classification . Primary 33C67;17B22. Secondary33D52 .
The theory of the hypergeometric functions associated to root systems started in the1980s with Heckman and Opdam via a generalization of the spherical functions onRiemannian symmetric spaces of noncompact type. Several important aspects arestudied by them in a series of publications [4, 9, 10, 11, 12]. One of the impressivedevelopments came in 1995s with the work of Opdam [11], where he introduced aremarkable family of orthogonal polynomials ( the so called Opdam’s nonsymmet-ric polynomials ) as simultaneous eigenfunctions of Cherednik operators. It containsin particular the presentation of the non-symmetric hypergeometric functions wheretheir investigations become an interesting topics in the theory of special functionsand in the harmonic analysis. In this paper, we focus on the non-symmetric hyperge-ometric function associated to root systems of type A , for the purpose in finding anexplicit formula for it, as it is done in the symmetric case [1, 2, 13, 3]. The setting isthat non-symmetric hypergeometric function can be derived from the symmeric onesvia application of a suitable polynomial of Cherednik operators ( [12], cor. 7.6 ).This paper deals with the case where the root system is of type A , using Opdam’sshifted operators and Cherednik operators, so the problem semble to be more robustfor others A n .In order to describe our approach let us be more specific about A -type hyperge-ometric function. We assume that the reader is familiar with root systems and theirbasic properties. As a general reference, we mention Opdam [11, 12].1 B. Amri and M. Bedhiafi
Let ( e , e , ..., e n ) be the standard basis of R n and h ., . i be the usual inner productfor which this basis is orthonormal. We denote by k . k its Euclidean norm. Let V bethe hyperplane orthogonal to the vector e = e + ... + e n . In V we consider the rootsystem of type A n − R = { e i − e j , ≤ i = j ≤ n } with the subsystem of positives roots R + = { e i − e j , ≤ i < j ≤ n } . The associated Weyl group W is isomorphic to symmetric group S n , permuting the n coordinates. We define the positive Weyl chamber C = { x ∈ V , x > x > ... > x n } . Denote π n the orthogonal projection onto V , which is given by π n ( x ) = x − n n X j =1 x j ! e, x ∈ R n . The cone of dominant weights is the set P + = n − X j =1 Z + β j , β j = π n ( e + e + ... + e j ) . For fixed k > , the Dunkl-Cherednik operators T ξ , ξ ∈ R n , is defined by T kξ = ∂ ξ + k X i 0) = 1 , for all symmetric polynomial p ∈ C [ X , .., X n ] . In particular ∆ k F k ( λ, . ) = k λ k F k ( λ, . ) (1.2)where ∆ k = P ni =1 (cid:0) T π ( e i ) (cid:1) is the Heckman-Opdam Laplacian. Note that the restric-tion of ∆ k to the set of W-invariant functions is the differential operator L k = ∆ + k X α ∈ R + coth h α, x i ∂ α + h ρ k , ρ k i onsymmetric Opdam’s hypergeometric function of type A ∆ is the ordinary Laplace operator.There exists a unique solution G k ( λ, x ) of the eigenvalue problem T ξ ( k ) G k ( λ, . ) = h λ, ξ i G k ( λ, . ) , ∀ ξ ∈ R N , G k ( λ, 0) = 1 . (1.3)holomorphic for all λ and for x in V + iU for a neighbourhood U ⊂ V of zero. Thefunction G k is the so-called nonsymmetric Opdam’s hypergeometric function.If x ∈ V then | G k ( λ, x ) | ≤ √ n ! e max w ∈ W h Re ( λ ) ,w ( x ) i . (1.4)Moreover, the Heckman-Opdam hypergeometric function F k can be written as F k ( λ, x ) = 1 n ! X w ∈ W G k ( λ, w.x ) . (1.5)In other words, for λ satisfying λ i − λ j = 0; ± k we have G k ( λ, x ) = D q F k ( λ, x ) , x ∈ V (1.6)where D q = Y ≤ i For all λ ∈ V C and x ∈ C . F k,n ( λ, x ) =Γ( nk )Γ( k ) n V n ( x ) k − Z x x ... Z x n − x n F k,n − (cid:16) π n − ( λ ) , π n − ( ν ) (cid:17) e | ν | ( − nk/ | λ | / ( n − ) V n − ( ν ) W k ( x, ν ) dν (2.2)In the rank-one case, which corresponds to take n = 2 and V = R ( e − e ) , theformula (2.2) becomes F k, ( λ, x ) = Γ(2 k )Γ( k ) (2 sinh x ) k − Z x − x e ν (1 − k +2 λ ) ( e x − e ν ) k − ( e ν − e − x ) k − dν = Γ(2 k )2 k Γ( k ) (sinh x ) k − Z x − x e νλ (cosh x − cosh ν ) k − dν = ϕ k − / , − / iλ ( x ) = F (cid:18) k − λ , k λ , k + 12 , − sinh x (cid:19) where ϕ k − / , − / iλ is a Jacobi function see (1.4), (3.4) and (3.5) of [7]. We recall herevarious facts about the Jacobi function ϕ k − / , − / iη that we shall need later, we referto [7, 8]. ϕ k − / , − / iη (2 t ) = ϕ k − / ,k − / iη ( t ) (2.3) (cid:16) ϕ k − / , − / iη (cid:17) ′ ( t ) = 1(2 k + 1) ( η − k ) sinh( t ) ϕ k +1 / , − / iη ( t ) (2.4) (cid:16) ϕ k − / , − / iη (cid:17) ′′ ( t ) + 2 k coth( t ) (cid:16) ϕ k − / , − / iη (cid:17) ′ ( t ) = ( η − k ) ϕ k +1 / , − / iη ( t ) (2.5)In the rank-two case, where n = 3 which is our subject in the next section,Heckman-Opdam’s hypergeometric function has the following integral representation( we omit here the dependance on n ) F k ( λ, x ) = Γ(3 k )Γ( k ) V ( x ) − k +1 Z x x Z x x e (1 − ( λ + k ))( ν + ν ) ϕ k − , − i ( λ − λ (cid:18) ν − ν (cid:19) ( e ν − e ν ) W k ( ν, x ) dν. (2.6)where W k ( ν, x ) = (cid:16) ( e x − e ν )( e x − e ν )( e x − e ν )( e ν − e x )( e ν − e x )( e ν − e x )) (cid:17) k − and V ( x ) = ( e x − e x )( e x − e x )( e x − e x ) . In order to find an expression for F k of Laplace type, we write F k ( λ, x ) = Γ(3 k )4Γ( k ) (cid:16)Q ≤ i 2; 1 ≤ j ≤ sinh (cid:18) | ν i − x j | (cid:19)! k − dν. B. Amri and M. Bedhiafi With the change of variables y = ν + ν , t = ν − ν , we have that F k ( λ, x ) = Γ(3 k )2 k − Γ( k ) (cid:16)Q ≤ i For f ∈ C ∞ c ( V ) and x ∈ C , we have the following intertwiningproperty ∆ k ( V k ( f )) = V k (∆( f )) . Proof. By inversion formula for Fourier transform and Fubini’s Theorem V k ( f )( x ) = Z R Z co ( x ) b f ( ξ ) e i h ξ,z i N k ( x, z ) dzdξ = Z R b f ( ξ ) F k ( iξ, x ) dξ. (2.9)Here we define the Fourier transform of f by b f ( ξ ) = 12 π Z R f ( t ) e − i h ξ,t i dt From a general estimates of Heckman-Opdam’s hypergeometric function ( see forexample corollary 6.2 in [12]), the last integral of (2.9) is a C ∞ as a function of x andthen by (1.2) one has ∆ k ( V k ( f ))( x ) = − Z R k ξ k b f ( ξ ) F k ( iξ, x ) dξ = Z R c ∆ f ( ξ ) F k ( iξ, x ) dξ = V k (∆( f )( x ) , which proves the desired fact. A We begin with some backgrounds from Heckman-Opdam theory of hypergeometricfunctions associated to a root system R with Weyl group W of a finite-dimensionalvector space a , we refer to [11, 12] for a more detailed treatment. For a regularweight λ ∈ P + (the set of dominant weights) we denote by P k ( λ, . ) the Hekman-Opdam Jacobi polynomial and by E k ( λ, . ) the non symmetric Opdam polynomial. Inthe following we collect some properties and relationships.(i) P k ( λ, x ) = X w ∈ W E k ( λ, wx ) (ii) X w ∈ W det ( w ) E k ( λ + δ, wx ) = V ( x ) P k +1 ( λ, x ) , where δ = 12 X α ∈ R + α and V ( x ) = Y α ∈ R + ( e α/ − e − α/ ) B. Amri and M. Bedhiafi (iii) P k ( λ, 0) = c k ( ρ k ) c k ( λ + ρ k ) , where ρ k ( λ ) = 12 X α ∈ R + k α α , c k ( λ ) = Y α ∈ R + Γ( h λ, ˇ α i )Γ( h λ, ˇ α i + k α ) and ˇ α = 2 α/ | α | .(iv) P k ( λ, x ) = P k ( λ, F k ( λ + ρ k , x ) (v) E k ( λ, x ) = P k ( λ, | W | G k ( λ + ρ k , x ) .From these facts we give the following consequence. Proposition 3.1. For all λ, x ∈ a , X w ∈ W det ( w ) G k ( λ, w.x ) = d k ( λ ) V ( x ) F k +1 ( λ, x ) , (3.1) where, d k ( λ ) = | W | c k +1 ( ρ k +1 ) c k ( ρ k ) c k ( λ ) c k +1 ( λ ) . Proof. This is closely related to ( ii ) , where by using the above relations one can writefor regular λ ∈ P + , X w ∈ W det ( w ) G k ( λ + ρ k +1 , x ) = d k ( λ + ρ k +1 ) V ( x ) F k +1 ( λ + ρ k +1 , x ) , since we have ρ k + δ = ρ k +1 . This identity can be extended for all λ ∈ a via analyticcontinuation by means of Carlson’s theorem.Let us return now to the space V , for n = 3 and investigate (3.1). In this case wehave d k ( λ ) = 1(2 k + 1)(3 k + 1)(3 k + 2) ( λ − λ + k )( λ − λ + k )( λ − λ + k ) . We introduce the antisymmetric function F ∗ k ( λ, x ) = 16 X w ∈ S det ( w ) G k ( λ, x ) = d k ( λ )6 V ( x ) F k +1 ( λ, x ) (3.2)and the following notations: I k ( λ, ν ) = ϕ k − , − i ( λ − λ (cid:18) ν − ν (cid:19) e − ( λ + k )( ν + ν ) ; γ k = Γ(3 k )Γ( k ) L k ( λ, ν ) = (cid:16) ϕ k − , − i ( λ − λ (cid:17) ′ (cid:0) ν − ν (cid:1) λ − λ − k e − ( λ + k )( ν + ν ) . So we can write F k ( λ, x ) = γ k V ( x ) − k +1 Z x x Z x x I k ( λ, ν ) e ν + ν ( e ν − e ν ) W k ( ν, x ) dν. (3.3)The integral representation for the functions F ∗ k is given in the following. onsymmetric Opdam’s hypergeometric function of type A Proposition 3.2. We have for λ ∈ V C and x ∈ C , F ∗ k ( λ, x ) = γ k k V ( x ) − k Z x x Z x x L k ( λ, ν ) p ( x, ν ) W k ( ν, x ) dν. (3.4) where p ( x, ν ) = − be ν + ν ) + ( ab + 3) e ν + ν ( e ν + e ν ) − a ( e ν + e ν ) − b + a ) e ν + ν + 4 b ( e ν + e ν ) − and with a = e x + e x + e x and b = e − x + e − x + e − x .Proof. We first write ( λ − λ + k )( λ − λ + k )( λ − λ + k )= ( λ − λ + k )4 (cid:16) ( − λ + k ) + 2 k ( − λ + k ) − (( λ − λ ) − k ) (cid:17) . The formula (2.6) for parameter k + 1 , together with (2.4) yield ( λ − λ + k ) F k +1 ( λ, x ) = 2(2 k + 1) γ k +1 V ( x ) − k − Z x x Z x x L k ( λ, ν ) W k +1 ( ν, x ) dν. So using integration by parts, ( − λ + k )( λ − λ + k ) F k +1 ( λ, x )= 2(2 k + 1) γ k +1 V ( x ) − k − Z x x Z x x L k ( λ, ν )(4 k − ( ∂ ν + ∂ ν )) W k +1 ( ν, x ) dν. and ( − λ + k ) ( λ − λ + k ) F k +1 ( λ, x )= 2(2 k + 1) γ k +1 V ( x ) − k − Z x x Z x x L k ( λ, ν )( ∂ ν + ∂ ν − k ) W k +1 ( ν, x ) dν. B. Amri and M. Bedhiafi In the same way with the use of (2.5), (( λ − λ ) − k )( λ − λ + k ) F k +1 ( λ, x )= − k + 1) γ k +1 V ( x ) − k − Z x x Z x x e − ( λ + k )( ν + ν ) ( λ − λ + k ) ϕ k − , − i ( λ − λ (cid:18) ν − ν (cid:19) ( ∂ ν − ∂ ν ) W k +1 ( ν, x ) dν = 2(2 k + 1) γ k +1 V ( x ) − k − Z x x Z x x e − ( λ + k )( ν + ν ) (cid:16) ϕ k − , − i ( λ − λ (cid:17) ′ (cid:0) ν − ν (cid:1) λ − λ − k ( ∂ ν − ∂ ν ) W k +1 ( ν, x ) dν − k (2 k + 1) γ k +1 V ( x ) − k − Z x x Z x x e − ( λ + k )( ν + ν ) (cid:16) ϕ k − , − i ( λ − λ (cid:17) ′ (cid:0) ν − ν (cid:1) λ − λ − k (cid:18) e ν + e ν e ν − e ν (cid:19) ( ∂ ν − ∂ ν ) W k +1 ( ν, x ) dν = 2(2 k + 1) γ k +1 V ( x ) − k − Z x x Z x x L k ( λ, ν ) n ( ∂ ν − ∂ ν ) − k (cid:18) e ν + e ν e ν − e ν (cid:19) ( ∂ ν − ∂ ν ) o W k +1 ( ν, x ) dν. Hence it follows that F ∗ k ( λ, x )= γ k k V ( x ) − k Z x x Z x x L k ( λ, ν ) n k + 2 k (cid:18) e ν + e ν e ν − e ν (cid:19) ( ∂ ν − ∂ ν ) − k ( ∂ ν + ∂ ν ) + 4 ∂ ν ∂ ν o W k +1 ( ν, x ) dν. Now, it is straightforward computation to verify that p ( ν, x ) = n k + 2 k (cid:18) e ν + e ν e ν − e ν (cid:19) ( ∂ ν − ∂ ν ) − k ( ∂ ν + ∂ ν ) + 4 ∂ ν ∂ ν o W k +1 ( ν, x ) Next, we can provide an integral expansion for G k by differentiating (3.3) and(3.4). Let us introduce the operators D k = D k ( λ ) = ( λ − λ + 2 k ) T π ( e ) + ( λ − λ + k ) T π ( e ) + τ ( λ ) + k ( λ − λ ) + k D ∗ k = D ∗ k ( λ ) = ( λ − λ − k ) T π ( e ) + ( λ − λ − k ) T π ( e ) + τ ( λ ) − k ( λ − λ ) + k where τ ( λ ) = λ + λ + λ λ . Recall that π ( e ) = ( , − , − ) and π ( e ) = ( − , , − ) .Our main result from this section is the following. Theorem 3.3. For λ, x ∈ V we have ( τ ( λ ) − k ) G k ( λ, x ) = D k ( F k ( λ, x )) + D ∗ k ( F ∗ k ( λ, x )) . (3.5) onsymmetric Opdam’s hypergeometric function of type A Proof. The proof is purely computational. Using the following fact, see (5.1 ) of [11], wT ξ w − = T w.ξ − X α ∈ R + , w.α ∈ R − k h α, ξ i s wα . we obtain T π ( e ) ( G k ( λ, x )) = λ G k ( λ, x ) ,T π ( e ) ( G k ( λ, s , .x )) = λ G k ( λ, s , .x ) − kG k ( λ, x ) ,T π ( e ) ( G k ( λ, s , .x )) = λ G k ( λ, s , .x ) ,T π ( e ) ( G k ( λ, s , .x )) = λ G k ( λ, s , .x ) − kG k ( λ, σ.x ) − kG k ( λ, x ) ,T π ( e ) ( G k ( λ, σ.x )) = λ G k ( λ, σ.x )) − kG k ( λ, s , .x ) T π ( e ) ( G k ( λ, σ .x )) = λ G k ( λ, σ .x )) − kG k ( λ, s , .x ) − kG k ( λ, s , .x ) , and T π ( e ) ( G k ( λ, x )) = λ G k ( λ, x ) ,T π ( e ) ( G k ( λ, s , .x )) = λ G k ( λ, s , .x ) + kG k ( λ, x ) ,T π ( e ) ( G k ( λ, s , .x )) = λ G k ( λ, s , .x ) − kG k ( λ, x ) T π ( e ) ( G k ( λ, s , .x )) = λ G k ( λ, s , .x ) − kG k ( λ, σ .x ) + kG k ( λ, σ.x ) ,T π ( e ) ( G k ( λ, σ.x )) = λ G k ( λ, σ.x )) − kG k ( λ, s , .x ) ,T π ( e ) ( G k ( λ, σ .x )) = λ G k ( λ, σ .x )) + kG k ( λ, s , .x ) . where σ = s , s , . Thus we have T π ( e ) ( F k ( λ, x )) = 16 n ( λ − k ) G k ( λ, x ) + ( λ − k ) G k ( λ, s , .x )+( λ − k ) G k ( λ, s , .x ) + λ G k ( λ, s , .x )+( λ − k ) G k ( λ, σ.x ) + λ G k ( λ, σ .x ) o T π ( e ) ( F k ( λ, x )) = 16 n λ G k ( λ, x ) + ( λ − k ) G k ( λ, s , .x )+( λ + k ) G k ( λ, s , .x ) + λ G k ( λ, s , .x )+( λ + k ) G k ( λ, σ.x ) + ( λ − k ) G k ( λ, σ .x ) o ,T π ( e ) ( F ∗ k ( λ, x )) = 16 n ( λ + 2 k ) G k ( λ, x ) − ( λ + k ) G k ( λ, s , .x ) − ( λ + 2 k ) G k ( λ, s , .x ) − λ G k ( λ, s , .x )+( λ + k ) G k ( λ, σ.x ) + λ G k ( λ, σ .x ) o , and T π ( e ) ( F ∗ k ( λ, x )) = 16 n λ G k ( λ, x ) − ( λ + k ) G k ( λ, s , .x )+( − λ + k ) G k ( λ, s , .x ) − λ G k ( λ, s , .x )+( λ − k ) G k ( λ, σ.x ) + ( λ + k ) G k ( λ, σ .x ) o . So, formula (3.5) can be checked by a straightforward calculations.2 B. Amri and M. Bedhiafi References [1] B. Amri, Note on the Bessel function of type A N − . 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