T. H. Baker and P. J. Forrester, The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), 175-216.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Laguerre polynomials on R . The latter two are related to the CalogeroSutherland models associated to the Weyl groups of type A and type B. 1. Introduction Recently the Calogero Sutherland models associated to the Weyl groups of type A and type B are studied by several authors (cf. [2, 3 ,4, 7, 12, 13, 15, 17] and the references there) The model of type A is the Calogero model with harmonic term, which consists of d particles on the circle with inverse square interaction in an harmonic potential, whose Hamiltonian is, after changing variable x k = exp(i k ) and conjugation with the ground state, ....

T. H. Baker and P. J. Forrester, The Calogero-Sutherland model and generalized classical polynomials, Commun. Math. Phys. 188 (1997), 175-216.

....(68) is given by (67) as (1; 1) 2 S n (a; b r; c) S n (a; b; c) 72) This formula is needed to calculate the degenerate cases of Kaneko s integral corresponding to Laguerre and Hermite polynomials. The generalized Laguerre polynomials L (y; are de ned by [23] see also [2]) y 1 ; y r ; lim ; 73) we use there the convention of [42] Let LS n (a; c) denote the Laguerre Selberg integral (26) One can deduce from (67) the Laguerre version of Kaneko s integral. Indeed, Kaneko s formula can also be written as [15] R(x; ....

.... R n dx i = 1) 2 kn(n 1) nr (kj) H (n ; i (79) where the generalized Hermite polynomials H (y; are de ned by H (y 1 ; y r ; lim 2a) j j (a y 1 2a; a y r 2a; 80) We follow here the convention of [2]. Kaneko s identity can be interpreted as a generalization of Heine s integral representation of orthogonal polynomials in the Jacobi case. Indeed, it can be rewritten as (t 1 ; t r ) 1;1] d t (x 1 ) d t (x n ) 81) where d t (x) Q r j=1 (t j x) 1 x) 1 ....

T H Baker and P J Forrester, The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), 175-216.

....not exists for other series of root systems, see [Ok10] 3 Also note, that by virtue of the binomial formula (2.6) the relation (1.21) is equivalent to formula (4.2) in [OO2] for the Bessel functions. Apparently, that formula for the Bessel functions was also known to M. Lassalle, see the paper [BF], Sections 3 and 6, where the authors cite Lassalle s unpublished letters. Lassalle s argument is very different from proofs of (1.21) given in [KS] and [Ok8] A proof and an explanation of (1.22) based on some general log concavity results was proposed in the paper [Ok3] See also the subsequent ....

T. H. Baker and P. J. Forrester, The Calogero-Sutherland model and generalized classical polynomials, q-alg/9608004.

....for the operators P N i=1 d i k (resp. e 2h P N j=1 x 2 j 1 P N i=1 ( 1 i k ) 1 e 0 2h P N j=1 x 2 j ) k = 1; N) They are labeled by the partitions and normalized so that the coefficient of the highest term (for the dominance ordering of the partitions) is 1 [19, 2]. The symmetric Jack (resp. Hermite) polynomials are written by the sum of the non symmetric Jack (resp. Hermite) polynomials as follows. J m (x) X oe2S m c oe 8 m oe (x) resp. H m (x) X oe2S m c oe 8 m(H) oe (x) 4.24) Remark that the coefficients c oe for the Jack case ....

....m i 0m j 0i j 1 ff 1 0 0 m i 0m j 1 0i j01 ff 1 0 0 m i 0m j 1 0i j ff 1 0 0 m i 0 m j 0i j ff 1 : Q (i;j)2m means the product over the (i; j) boxes contained in the Young diagram m. To get the formula (4. 29) we used the following formulas [16] chap.VI 10.38, [2] prop 3.7. hJ m (x) J m (x)i J = Y 1i jN 0 0 m i 0 m j 0i j 1 ff 1 0 0 m i 0 m j 1 0i j01 ff 1 0 0 m i 0m j 1 0i j ff 1 0 0 m i 0m j 0i j ff 1 ; 4.30) N Y i=1 Z 1 01 e 0x 2 i dx i Y j k jx j 0 x k j 2 ff = 2 0 N(N01) 2ff N 2 ....

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T. H. Baker, and P. J. Forrester,: The Calogero-Sutherland model and generalized classical polynomials, Preprint RIMS-1094 (1996). (solv-int/9608004)

....group SN , which were first introduced by Lassalle in [L2] Moreover, the generalized Laguerre polynomials of [B F2] B F3] which are non symmetric analogues of those in [L1] can be considered as a subsystem of Hermite polynomials associated with a reflection group of type BN . We refer to [B F1] and [vD] for a thorough study of the symmetric multivariable Hermite and Laguerre systems. After a short collection of notations and basic facts from Dunkl s theory in Section 2, the concept of generalized Hermite polynomials is introduced in Section 3, along with a discussion of the above ....

Baker, T.H., Forrester, P.J.: The Calogero-Sutherland model and generalized classical polynomials. Comm. Math. Phys. 188 175--216 (1997).

....series 0 F 0 (x; y) 33] which is used to define generalizations of the Fourier and Laplace transforms. This kernel also allows us to derive an exponential formula, a generating function, and integral formulae for the nonsymmetric Hermite polynomials in complete analogy with the symmetric case [1]. Indeed, we show that 0 F 0 (x; y) can be constructed from KA (x; y) by symmetrization. The analysis of Section 3 is repeated, albeit more succinctly, in Section 4 for the Laguerre case. 2 The Jack case We begin by reviewing some of the results in [18, 29] A fundamental result concerns the ....

....multiplied by n gives the sought result. 2 The hypergeometric function 0 F 0 (x; y) is related to the symmetric Hermite polynomials by a generating function analogous to Proposition 3. 9, and also through an integral transform of the symmetric Jack polynomials, in which 0 F 0 (x; y) is the kernel [19, 1]. Likewise, KA (x; y) also occurs as the kernel in an integral transform which relates the non symmetric Jack and Hermite polynomials. Gammay jy j Gamma y k j 15 Then we have KA (2y; z)KA (2y; w) d p2 (w) p2 (z) KA (2z; w) 3.28) KA (2y; z) E j (y) d E j (z) ....

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T. H. Baker and P. J. Forrester. The Calogero--Sutherland model and generalized classical polynomials. solv-int/9608004.

....over all partitions which have the same modulus as but are smaller in dominance ordering. The polynomials P possess a host of special properties, and in fact form the natural basis for a class of symmetric multivariable orthogonal polynomials generalizing the classical orthogonal polynomials [7, 8, 2]. Although through the efforts of Macdonald [9] Stanley [12] and others, the theory of symmetric Jack polynomials is highly developed, many theorems seem difficult to prove. One reason for this is that the symmetric Jack polynomials are not the most fundamental polynomials in the theory this ....

T. H. Baker and P. J. Forrester, "The Calogero--Sutherland model and generalized classical polynomials ", solv-int/9608004, to appear in Comm. Math. Phys.

....(OE j Gamma OE k ) OE j OE k ) 1. 12) respectively, where the operator S j replaces the coordinate x j (OE j ) by Gammax j ( GammaOE j ) the superscripts (H) L) and (J) stand for Hermite, Laguerre and Jacobi due to the relationship with these classical polynomials in the case N = 1 [7]) The symmetric ground state eigenfunctions of (1.10) and (1.11) are of the form e with W given by 1 log jx k Gamma x j j (1.13) 1 log x log jx j j; 1.14) log sin b log cos log j sin OE j Gamma sin OE k j: 1.15) Also, the ....

....we can construct the unique eigenfunction of the form c j E (y; ff) 3.2) again with eigenvalue Gamma2jjj. We will refer to the E j (y; ff) as the non symmetric generalized Hermite polynomials (they are related to the symmetric generalized Hermite polynomials defined in [7] by an equation analogous to (2.14) see eq. 3.16) below) In fact by adopting a method due to Sogo [18] an exponential operator formula can be obtained expressing E j (y; ff) in terms of E j (y; ff) which is the analogue of the formula due to Lassalle [19] see eq. 3.16) below) expressing ....

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T. H. Baker and P. J. Forrester. The Calogero--Sutherland model and generalized classical polynomials. solv-int/9608004.

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T. H. Baker and P. J. Forrester, The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), 175-216.

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