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MARTIN HALLNÄS AND EDWIN LANGMANNAbstract. In this paper we consider a large class of many-variable polynomials which contains generalisations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogero-Sutherland type operators and their deformations recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present a unified and explicit construction of all these polynomials.
We obtain kernel functions associated with the quantum relativistic Toda systems, both for the periodic version and for the nonperiodic version with its dual. This involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic Calogero-Moser systems. We show that the special kernel functions at issue admit a limit that yields generating functions of Bäcklund transformations for the classical relativistic Calogero-Moser and Toda systems. We also obtain the nonrelativistic counterparts of our results, which tie in with previous results in the literature.
In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions J N (a + , a − , b; x, y) of the Hamiltonians arising in the integrable N -particle systems of hyperbolic relativistic Calogero-Moser type. We focused on the first steps of the scheme in Part I, and on the cases N = 2 and N = 3 in Part II. In this paper, we determine the dominant asymptotics of a similarity transformed function E N (b; x, y) for y j − y j+1 → ∞, j = 1, . . . , N −1, and thereby confirm the long standing conjecture that the particles in the hyperbolic relativistic Calogero-Moser system exhibit soliton scattering. This result generalizes a main result in Part II to all particle numbers N > 3
Abstract. We introduce and study natural generalisations of the Hermite and Laguerre polynomials in the ring of symmetric functions as eigenfunctions of infinite-dimensional analogues of partial differential operators of Calogero-Moser-Sutherland (CMS) type. In particular, we obtain generating functions, duality relations, limit transitions from Jacobi symmetric functions, and Pieri formulae, as well as the integrability of the corresponding operators. We also determine all ideals in the ring of symmetric functions that are spanned by either Hermite or Laguerre symmetric functions, and by restriction of the corresponding infinite-dimensional CMS operators onto quotient rings given by such ideals we obtain socalled deformed CMS operators. As a consequence of this restriction procedure, we deduce, in particular, infinite sets of polynomial eigenfunctions, which we shall refer to as super Hermite and super Laguerre polynomials, as well as the integrability, of these deformed CMS operators. We also introduce and study series of a generalised hypergeometric type, in the context of both symmetric functions and 'super' polynomials.
We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials SP λ ((z1, . . . , zn), (w1, . . . , wm); θ) with respect to a natural positive semi-definite, but degenerate, Hermitian product ·, · n,m,θ . In case m = 0 (or n = 0), our product reduces to Macdonald's well-known inner product ·, · n,θ , and we recover his corresponding orthogonality results for the Jack polynomials P λ ((z1, . . . , zn); θ). From our main results, we readily infer that the kernel of ·, · n,m,θ is spanned by the super-Jack polynomials indexed by a partition λ not containing the m × n rectangle (m n ). As an application, we provide a Hilbert space interpretation of the deformed trigonometric Calogero-Moser-Sutherland operators of type A(n − 1, m − 1).
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