279 research outputs found

    A generalized Macdonald operator

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    We present an explicit difference operator diagonalized by the Macdonald polynomials associated with an (arbitrary) admissible pair of irreducible reduced crystallographic root systems. By the duality symmetry, this gives rise to an explicit Pieri formula for the Macdonald polynomials in question. The simplest examples of our construction recover Macdonald's celebrated difference operators and associated Pieri formulas pertaining to the minuscule and quasi-minuscule weights. As further by-products, explicit expansions and Littlewood-Richardson type formulas are obtained for the Macdonald polynomials associated with a special class of small weights.Comment: 11 pages. To appear in Int. Math. Res. Not. IMR

    On certain multiple Bailey, Rogers and Dougall type summation formulas

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    A multidimensional generalization of Bailey's very-well-poised bilateral basic hypergeometric 6ψ6{}_6\psi_6 summation formula and its Dougall type 5H5{}_5H_5 hypergeometric degeneration for q→1q\to 1 is studied. The multiple Bailey sum amounts to an extension corresponding to the case of a nonreduced root system of certain summation identities associated to the reduced root systems that were recently conjectured by Aomoto and Ito and proved by Macdonald. By truncation, we obtain multidimensional analogues of the very-well-poised unilateral (basic) hypergeometric Rogers 6ϕ5{}_6\phi_5 and Dougall 5F4{}_5F_4 sums (both nonterminating and terminating). The terminating sums may be used to arrive at product formulas for the norms of recently introduced (qq-)Racah polynomials in several variables.Comment: 20 pages, LaTe

    Remarks on the Zeros of the Associated Legendre Functions with Integral Degree

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    We present some formulas for the computation of the zeros of the integral-degree associated Legendre functions with respect to the order.Comment: 7 pages, 2 figure

    Properties of some families of hypergeometric orthogonal polynomials in several variables

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    Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the norms of the polynomials in terms of the norm of the constant polynomial.Comment: 42 pages, AMSLaTeX 1.1 with amssym

    Scattering theory of discrete (pseudo) Laplacians on a Weyl chamber

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    To a crystallographic root system we associate a system of multivariate orthogonal polynomials diagonalizing an integrable system of discrete pseudo Laplacians on the Weyl chamber. We develop the time-dependent scattering theory for these discrete pseudo Laplacians and determine the corresponding wave operators and scattering operators in closed form. As an application, we describe the scattering behavior of certain hyperbolic Ruijsenaars-Schneider type lattice Calogero-Moser models associated with the Macdonald polynomials.Comment: 31 pages, LaTe

    On the Equilibrium Configuration of the BC-type Ruijsenaars-Schneider System

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    It is shown that the ground-state equilibrium configurations of the trigonometric BC-type Ruijsenaars-Schneider systems are given by the zeros of Askey-Wilson polynomials.Comment: 7 pages, LaTe

    Self-dual Koornwinder-Macdonald polynomials

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    We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verify Macdonald's normalization conjectures for these polynomials.Comment: 21 pages, AMSLaTeX 1.1 with amssym
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