DIFFERENTIAL EQUATIONS AND RECURSION RELATIONS FOR LAGUERRE FUNCTIONS ON SYMMETRIC CONES

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volue 359, Nuber 7, July 2007, Pages S (07) Article electronically published on February 8, 2007 DIFFERENTIAL EQUATIONS AND RECURSION RELATIONS FOR LAGUERRE FUNCTIONS ON SYMMETRIC CONES HONGMING DING Abstract. We obtain the differential equation recurrence relations satisfied by the Laguerre functions l on an arbitrary syetric cone. 1. Introduction It is well known that the classical Laguerre polynoials L α n for α> 1 aybe defined [1] by n ( (1.1) L α n 1 n(x) =(α +1) n ( x) k) k, (α +1) k k=0 they for an orthogonal basis for L 2 (R +,e x x α dx). It follows that the classical Laguerre functions ln α defined by (1.2) ln(x) α =e x L α n(2x) are an orthogonal basis for L 2 (R +,x α dx). Moreover, they satisfy the differential equation (1.3) x d2 ln(x) α dx 2 +(α +1) dlα n(x) xl α dx n(x) = (2n + α +1)ln(x), α the following recurrence relations: x d2 ln(x) α (1.4) dx 2 +(2x + α +1) dlα n(x) +(x + α +1)l α dx n(x) = 2(n + α)ln 1(x), α x d2 ln(x) α (1.5) dx 2 (2x α 1) dlα n(x) +(x α 1)ln α dx (x) = 2(n +1)lα n+1 (x). As in [2], we refer to the differential operators on the left of (1.4) (1.5) as the annihilation creation operators, respectively. By (1.4) (1.5), (1.6) 2x dlα n(x) +(α +1)l α dx n(x) =(n +1)ln+1(x) α (n + α)ln 1(x). α By (1.3) - (1.5), we have (1.7) xl α n(x) =(n + α +1 2 )l α n(x) n + α 2 ln 1(x) α n +1 l α 2 n+1(x). Received by the editors August 24, 2004, in revised for, May 2, Matheatics Subject Classification. Priary 33C45; Secondary 32M15. Key words phrases. Jordan algebra, syetric cone, spherical polynoial, Laguerre polynoial, Laguerre function, Laplace transfor, gradient, differential equation, recurrence relation c 2007 Aerican Matheatical Society Reverts to public doain 28 years fro publication

2 3240 HONGMING DING It is also well known that the definitions (1.1) (1.2) of the Laguerre polynoials Laguerre functions have been generalized to syetric cones (see (2.8) (2.9) below). [12] discusses the differential equations satisfied by the Laguerre polynoials Laguerre functions on the syetric cones of positive definite atrices over C R. Using an approach of Lie group representations, [4] generalizes (1.6) to an arbitrary syetric cone. Using coputations of atrices, [2] generalizes (1.3) - (1.5) to the cones of positive definite atrices over C, or positive definite Heritian atrices. Using the ethod of Jordan algebras, we generalize in this paper all relations (1.3) - (1.7) to all syetric cones, which include the cones of positive definite atrices over R, C, H, the Lorentz cones, the exceptional cone of 3 3 positive definite atrices over the Cayley algebra. Hence, this paper is a generalization of [2] [12] to a general setting. Moreover, our ethod is sipler, uses uch less notation than in [4] [2]. Finally, we also obtain the recurrence relations involving the first-order differential annihilation creation operators (see Theore 4.5 below). As [13] indicated, our ethod for syetric cones cannot be used for Laguerre polynoials with general ultiplicities d. Such a ore general case will be studied in the next paper. This paper is organized into four sections, as follows. In Section 2, we review the definitions structures of Jordan algebras syetric cones, the definition of Laguerre functions on these cones. In Section 3, we define the gradient of a C-valued function as well as a V-valued function f on a Euclidean Jordan algebra V. We review soe recurrence forulas for spherical polynoials Φ.We also obtain soe gradient forulas for soe functions. In Section 4, we obtain our ain results of this paper, the differential equations recursion forulas for Laguerre functions on syetric cones. 2. Laguerre functions on syetric cones In this section, we review the structure of Jordan algebras syetric cones, the definition of Laguerre functions on these cones, that are needed in this paper. We refer to [7] for details. Let V be a siple Euclidean Jordan algebra, denote by n its diension as a real vector space, denote its rank by r, lete be the identity eleent in V. The interior of the subset of all eleents x 2 where x V is an irreducible syetric cone. Any irreducible syetric cone is isoorphic to a cone of this kind, any syetric cone is the direct product of irreducible syetric cones. Fixing a Jordan frae {c 1,...,c r } in V,wehavee = r c j. Denoting by the Jordan product in V, V has the following subspaces: V j = {x V : c j x = x} V jk = {x V : c j x = 1 2 x c k x = 1 2 x}. Then V j = Rc j for j =1,...,r are 1-diensional subalgebras of V, while the subspaces V jk for j, k =1,...,r with j<kall have a coon diension d. Then, V has the Pierce decoposition ( ) ( (2.1) V = V j V jk ), which is the orthogonal direct su. It follows that n = r + d 2 r(r 1). The trace in V is the linear functional tr x = x e, j<k

3 LAGUERRE FUNCTIONS 3241 where is the inner product in V. The characteristic function ψ of is defined by ψ(x) = e x y dy for all x, the Koecher nor function is given by (x) =cψ(x) r n, where c is a constant deterined by the noralization (e) =1. If (x) 0, then x is invertible, there is a V -valued polynoial Q of degree r 1 such that x 1 = (x) 1 Q(x). Let G be the connected coponent of the identity in the autoorphis group G(). Then G acts transitively on, = G/K where K is the stability group of the identity eleent e in ; i.e., K consists of all k G such that k e = e. For any x V,thereisk K ξ 1,...,ξ r R such that (2.2) x = k (ξ 1 c ξ r c r ). We refer to (2.2) as the polar decoposition of x. When x is written in the for (2.2), r (2.3) tr(x) = ξ r (x) = ξ r. For j =1,...,r,letE j = c c j,setj j = {x V : E j x = x}. Denote by P j the orthogonal projection of V onto the subalgebra J j, define j (x) =δ j (P j x) for x V,whereδ j denotes the Koecher nor function for J j. Then j is a polynoial on V that is hoogeneous of degree j. Letλ =(λ 1,...,λ r ) C r, define the function λ on V by r 1 (2.4) λ (x) = (x) λ r j (x) λ j λ j+1. In particular, when λ j = j are integers for all j =1,...,r (2.5) 1 r 0, =( 1,..., r ) is called a partition, (2.4) defines a polynoial function on V that is hoogeneous of degree = r. For each partition, the spherical polynoial of weight on V ay be defined by Φ (x) = (k x)dk. K It follows that Φ is a K-invariant hoogeneous polynoial of degree. Byanalytic continuation to the coplexification V C of V,tr,,Φ can be extended to polynoial functions on V C. The gaa function Γ for the cone is defined on C r by (2.6) Γ (λ) = e tr x λ (x) (x) n r dx whenever the integral converges absolutely. By [7, VII.1.1], in the range Re λ j > (j 1) d 2

4 3242 HONGMING DING of the variable λ, the integral (2.6) converges absolutely, Γ is evaluated in ters of the classical gaa function as r (2.7) Γ (λ) =(2π) 1 2 (n r) Γ(λ j (j 1) d 2 ). For λ C r any partition we define [λ] = Γ (λ + ). Γ (λ) For α C nonnegative integer j theclassicalpochhaersybol(α) j is defined by Γ(α + j) (α) j = = α(α +1) (α + j 1). Γ(α) It follows fro (2.7) that r ( [λ] = λ j (j 1) d ). 2 j The function Φ (e + x) isak-invariant polynoial of degree, hence has an expansion Φ (e + x) = ( ) Φ n n (x), n ( ) where is the generalized binoial coefficients. For R, the generalized n Laguerre polynoials are defined by (2.8) L (x) =[] n ( n ) 1 [] n Φ n ( x), the generalized Laguerre functions by (2.9) l(x) =e tr(x) L (2x). Let T = V + i be the Siegel upper half plane in V C. By [7, Proposition XV.4.2], for (2.10) >(r 1) d 2 z T, (2.11) e i z x l(x) (x) n r dx =Γ (+) (e iz) Φ ((z ie)(z +ie) 1 ), where (z + ie) 1 is the inverse of z + ie in V C. In this paper, we assue condition (2.10), which is also the assuption in [2], [4], [12]. In the classical case, (2.10) becoes α> 1, as discussed in the Introduction. By [7, XIII.1], define L 2 () = L 2 (, (2u) n r du), H 2 (T ) as the space of holoorphic functions F on T such that F 2 = F (z) 2 (y) 2n r dx dy <, T

5 LAGUERRE FUNCTIONS 3243 where z = x + iy. Iff belongs to L 2 (), then the function F, (2.12) F (z) =(2π) n 2 e i z s f(s) (2s) n r ds, belongs to H 2 (T ), the ap L : f F is a linear isoorphis fro L 2 () onto H 2 (T ). It follows fro (2.11) (2.12) that ( ) z + ie L l =[] Φ ((z ie)(z + ie) 1 ). 2i 3. The gradient of a function on V the recurrence relations for Φ In this section, we define the gradient for a C-valued a V-valued function f on a siple Euclidean Jordan algebra V, obtain soe gradient results for soe functions. We also review soe recurrence forulas for the spherical polynoials Φ. Let f : V R be a differentiable function; i.e., all directional derivatives D u,u V,exist.Fors V, we define the gradient f(s) V of f by the forula (3.1) f(s) u = D u f(s) = d dt f(s + tu) t=0. For a C-valued function f = f 1 + if 2, we define f = f 1 + i f 2.Forz = x + iy V C, we define D z = D x + id y. Let {e 1,...,e n } be an orthonoral basis of V, s = n s αe α V C,u = n u αe α V C. By (3.1), f(s) u = f(s) s α ū α, f(s) (3.2) f(s) = e α. s α It is easy to see that (3.2) is independent of the choice of an orthonoral basis {e 1,...,e n } of V so that (3.2) is an equivalent definition of f(s). Moreover, a function f : V V ay be expressed by (3.3) f(s) = f α (s)e α under an orthonoral basis {e 1,...,e n } of V. Define f by f α (s) (3.4) f(s) = e α e β. s β α,β=1 It is also easy to see that the right side of (3.4) is independent of the choice of an orthonoral basis {e 1,...,e n }, f(s) is well defined. Alternatively, one ay define f(s) as n f α (s) α,β=1 s β e α e β in the tensor product V V. However, we use (3.4) to keep f(s) V, a sipler space. By a siple coputation, the product rule of differentiation (3.5) tr( (f(s) g(s))) = tr(( f(s)) g(s)) + tr(f(s) g(s))

6 3244 HONGMING DING holds for V -valued functions f g, wherev is a Euclidean Jordan algebra. When f is a C-valued function, the product is the scalar product, (3.5) also holds. By (3.2) (3.4), if f is a C-valued function on V,then Lea 3.1. f(s) = α,β=1 (3.6) s = n r e. Proof. By (3.4) Since s α s β = δ αβ, s = (3.7) s = α,β=1 2 f(s) s α s β e α e β. s α s β e α e β. e α e α = e 2 α. Let {c 1,...,c r } be a Jordan frae of V decopose V into the orthogonal direct su (2.1). Then an orthonoral basis {e 1,...,e n } of V ay be fored by adding eleents in V jk to this Jordan frae. By [7, Proposition IV.1.4(i)], if e α V jk, then (3.8) e 2 α = e α e α = 1 2 (c j + c k ). It follows fro (3.7), (3.8), the fact c 2 j = c j that s = e 2 α = c j + n r r c j = n r c j = n r e. Lea 3.2. For R, an invertible eleent s V C, (3.9) ( (s) )= (s) s 1. Proof. If s V,thenshas the spectral decoposition; i.e., there is a Jordan frae {c 1,...,c r } s 1,...,s r R such that s = r s jc j. If s is invertible, then s j 0forj =1,...,r s 1 = r s 1 j c j. Wecopletethesystec 1,...,c r to an orthonoral basis {e α } by adding eleents belonging to V jk. Since (s) is K-invariant, it follows fro [7, Lea VI.4.3] that (s) s α =0ife α belongs to V jk (j<k). By (3.2) (2.3), ( (s) )= ( (s) ) s α e α = ( (s) ) s j c j = (s) s j c j = (s) s 1. Hence, the lea is proved for s V. By analytic continuation, the lea also holds for s V C.

7 LAGUERRE FUNCTIONS 3245 Lea 3.3. For R, an invertible eleent s V C, (3.10) (s (s) n r )= (s) n r e, (3.11) (s (s) n r )= ( (s) n r e)= ( (s) n r ). Proof. By (3.5), (3.6), (3.9), (s (s) n r )= (s) n r s + s ( (s) n r ) = (s) n n r r e + s ( n r ) (s) n r s 1 = (s) n r e, which is (3.10). (3.11) follows fro (3.10), (3.2), (3.4), the fact that e is the identity eleent of V. Let =( 1,..., r ) be a partition. Define (3.12) + γ j =( 1,..., j 1, j +1, j+1,..., r ), γ j =( 1,..., j 1, j 1, j+1,..., r ) whenever they do not violate condition (2.5) [14]. [10] coputed the binoial coefficient ( ) =( γ j + d j 2 (r j)) j k + d 2 (k j 1) k j j k + d 2 (k j). We adopt the notation (3.13) C (j) = k j k j d 2 (k +1 j) k j d 2 (k j), denote tr ( )f(s) =tr( f(s)), tr (s )f(s) =tr(s f(s)), tr (s 2 )f(s) = tr (s 2 f(s)). By (3.12), γ 1 =(1, 0,...,0). It is known that Φ γ1 (s) = 1 r tr(s). The following recurrence forulas for the spherical polynoials Φ,soeofwhich involve the gradient, can be found in [11]. Lea 3.4. Let be a partition s V C.Then, (3.14) tr (s)φ (s) = c (j)φ +γj (s), (3.15) tr( )Φ (s) = (3.16) tr (s 2 )Φ (s) = ( γ j ) Φ γj (s), ( j (j 1) d 2 ) c (j)φ +γj (s). Since tr (s ) is the Euler operator, Φ is a hoogeneous polynoial of degree on V C,wehave Φ (s) (3.17) tr(s )Φ (s) = s α = Φ (s). s α

8 3246 HONGMING DING 4. Differential equation recursion relations for l In this section, we generalize the ain result of [2], Theore 5.1, fro the syetric cones of Heritian atrices to arbitrary syetric cones. This generalization is carried out in Theores Theore 4.2 is a new recurrence relation, Theore 4.5 gives recurrence relations involving the first-order differential annihilation creation operators. Theore 4.1 recovers the ain result of [4], Theore 7.9, though our ethod is uch sipler. Theore 4.1. The Laguerre functions are related by the following recurrence relations: (4.1) tr (2s + e)l(s) = C (j)l+γ j (s) ( j 1+ (j 1) d ( ) 2 ) l γ γ j (s). j Proof. By (3.5), (3.10), the fact that every s is invertible, tr ( (sl(s) (s) n r )) =tr(s (s) n r l (s)) + l (s)tr( (s (s) n r )) = (s) n r tr (s + e) l (s) (4.2) (s) n r tr(2s + e) l (s) = tr (2 (sl (s) (s) n r )) r l (s) (s) n r. (2.11) ay be viewed as a Laplace transfor of l(x) (x) n r. Considering this transfor its inverse transfor [5, (3.1.1) (3.1.2)], it follows fro (4.2) that e i z s (s) n r tr (2s + e) l (s) ds (4.3) By (3.9), =tr( 2z e)(γ ( + ) (e iz) Φ ((z ie)(z + ie) 1 )). (4.4) tr (z ) (e iz) =tr(z ( ) (e iz) ( i)(e iz) 1 ) = 2 (e iz) tr (e +(z ie)(z + ie) 1 ). Denote (4.5) w =(z ie)(z + ie) 1, w as the gradient with respect to w. By (4.5), (3.1), a siple calculation, tr (z )Φ ((z ie)(z + ie) 1 ) = tr (2iz (z + ie) 2 w )Φ (w) (4.6) = 1 2 tr ((e w2 ) w )Φ (w).

9 LAGUERRE FUNCTIONS 3247 By (3.5), (4.4) - (4.6), (4.7) tr ( 2z e)( (e iz) Φ ((z ie)(z + ie) 1 )) =Φ ((z ie)(z + ie) 1 )tr( 2z ) (e iz) + (e iz) tr( 2z )Φ ((z ie)(z + ie) 1 ) r (e iz) Φ ((z ie)(z + ie) 1 ) = (e iz) tr (w +(w 2 e) w )Φ (w). By (4.3), (4.7), (3.14) - (3.16), we have e i z s (s) n r tr (2s + e) l (s) ds { ( ) =Γ ( + ) (e iz) Φ γ γj (w) j (4.8) By (2.7), + ( j + (j 1) d } 2 ) C (j)φ +γj (w). (4.9) Γ ( + ) =Γ ( + γ j )( j 1+ (j 1) d 2 ), (4.10) Γ ( + + γ j )=Γ ( + )( j + (j 1) d 2 ). By (4.8) - (4.10), the fact that L is a linear isoorphis fro L 2 () onto H 2 (T ), we have (4.1). Theore 4.2. The Laguerre functions are related by the following recurrence relations: (4.11) tr (s)l (s) =( + r 2 ) l (s) C (j) l+γ j (s) ( j 1+ (j 1) d ( ) 2 ) l γ γ j (s). j Proof. Siilar to (4.4) (4.6), we have (4.12) tr (i ) (e iz) = 2 (e iz) tr (w e), (4.13) tr (i )Φ ((z ie)(z + ie) 1 )= 1 2 tr ((e 2w + w2 ) w )Φ (w),

10 3248 HONGMING DING where w is given by (4.5). By (3.5), (3.14) - (3.17), (4.12), (4.13), itr ( )( (e iz) Φ ((z ie)(z + ie) 1 )) { = (e iz) ( + r 2 )Φ (w) 1 ( ) Φ 2 γ γj (w) j 1 ( j + (j 1) d } (4.14) 2 2 ) C (j)φ +γj (w). Siilar to (4.3), e i z s tr (s) l(s) (s) n r ds (4.15) = itr ( )(Γ ( + ) (e iz) Φ ((z ie)(z + ie) 1 )). By (4.9), (4.10), (4.14), (4.15), we have (4.11). Theore 4.3. TheLaguerrefunctionl satisfies the following differential equation: (4.16) tr ( s + s) l(s) =(r +2 ) l(s). Proof. By (3.5), (3.10), (3.11), we have tr ( (sl(s) (s) n r )) = (s) n r tr (s l (s)) + 2tr (( l(s)) ( (s (s) n r ))) + l(s)tr ( (s (s) n r )) (4.17) = (s) n r tr (s l (s)) + 2 (s) n r tr ( l (s)) + l(s)tr( ) (s) n r, tr ( (l(s) (s) n r )) =tr( (s) n r l (s)+l(s) ( (s) n r )) (4.18) = (s) n r tr ( l (s)) + l(s)tr( ) (s) n r. By (4.17) (4.18), tr ( (sl(s) (s) n r )+ (l (s) (s) n r )+sl (s) (s) n r ) = (s) n r tr ( s + s) l (4.19) (s). By (2.11) (4.19), e i z s (s) n r tr ( s + s) l (s) ds (4.20) = i tr ((z 2 + e) + z)(γ ( + ) (e iz) Φ ((z ie)(z + ie) 1 )). By (4.5), (3.1), a calculation, (z 2 + e) =2iw w, by (3.17), tr ((z 2 + e) )Φ ((z ie)(z + ie) 1 )=2itr(w w )Φ (w) (4.21) =2i Φ (w) =2i Φ ((z ie)(z + ie) 1 ). Siilar to (4.4), we have (4.22) tr ((z 2 + e) ) (e iz) = (e iz) tr (z)+ir (e iz).

11 LAGUERRE FUNCTIONS 3249 By (4.20) - (4.22), e i z s (s) n r tr ( s + s) l (s) ds =(r +2 )Γ ( + ) (e iz) Φ ((z ie)(z + ie) 1 ). Then the theore follows fro (2.11). Theore 4.4. The Laguerre functions are related by the following recurrence relations: 1 2 tr (s + +2s +s+e) l (s)= ( j 1+ (j 1) d ( ) 2 ) l γ γ j (s), j 1 2 tr ( s +2s s + e)l (s) = C (j)l+γ j (s). Proof. The theore follows directly fro (4.1), (4.11), (4.16), a siple coputation. Siilarly, it follows fro (4.1) (4.11) that Theore 4.5. The Laguerre functions are related by the following recurrence relations: tr (s + s r e) l (s) = ( j 1+ (j 1) d ( ) 2 ) l γ γ j (s), j tr (s s +( r + )e)l (s) = C (j)l+γ j (s). Acknowledgeent The author would like to thank M. Davidson, G. Ólafsson, G. Zhang for helpful discussions. References 1. G. Andrews, R. Askey, R. Roy, Special Functions, Cabridge Univ. Press, MR (2000g:33001) 2. M. Davidson G. Ólafsson, Differential Recursion Relations for Laguerre Functions on Heritian Matrices, Integral Transfor. Spec. Funct. 14 (6) (2003), MR (2004k:33017) 3. M. Davidson, G. Ólafsson, G. Zhang, Laguerre Polynoials, Restriction Principle, Holoorphic Representations of SL(2, R), Acta Appl. Math. 71 (2002), MR (2003f:22015) 4., Laplace Segal-Bergan Transfor on Heritian Syetric Spaces Orthogonal Polynoials, J.Funct.Anal.204 (1) (2003), MR (2004j:43012) 5. H. Ding K. Gross, Operator-valued Bessel Functions on Jordan Algebras,J.Reine.Angew. Math. 435 (1993), MR (93:33010) 6. H. Ding, K. Gross, D. Richards, Raanujan s Master Theore for Syetric Cones, Pacific J. Math. 175 (2) (1996), MR (98b:43019) 7. J. Faraut A. Koranyi, Analysis on Syetric Cones, Clarendon Press, Oxford, MR (98g:17031) 8. S. Helgason, Groups Geoetric Analysis, Acadeic Press, MR (86c:22017)

12 3250 HONGMING DING 9. B. Kostant, On Laguerre Polynoials, Bessel Functions, Hankel Transfor a Series in the Unitary Dual of the Siply-connected Covering Group of SL(2, R), Represent. Theory 4 (2000), MR (2001f:22046) 10. M. Lassalle, Coefficients Binoíaux Généralisés et Polynóes de Macdonald, J.Funct.Anal. 158 (1998), MR (2000a:33028) 11. B. Orsted G. Zhang, Generalized Principal Series Representations Tube Doains, Duke Math. J. 78 (1995), MR (96c:22015) 12. F. Ricci A. Vignati, Bergan Spaces on Soe Tube Type Doains Laguerre Operators on Syetric Cones, J. Reine. Angew. Math. 449 (1994), MR (95f:32042) 13. Z. Yan, Generalized Hypergeoetric Functions Laguerre Polynoials in Two Variables, Contep. Math. 138 (1992), MR (94j:33019) 14. G. Zhang, Soe Recurrence Forulas for Spherical Polynoials on Tube Doains, Trans. Aer. Math. Soc. 347 (5) (1995), MR (95h:22018) Departent of Matheatics Coputer Science, St. Louis University, St. Louis, Missouri E-ail address: dingh@slu.edu