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    • New Orthogonality Relations for Super-Jack Polynomials and an Associated Lassalle–Nekrasov Correspondence
    Authors: Martin, Hallnäs;  Advisor: -;  Co-Author: - (2023)

    The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in n+m variables, which reduce to the Jack polynomials when n=0 or m=0 and provide joint eigenfunctions of the quantum integrals of the deformed trigonometric Calogero–Moser–Sutherland system. We prove that the super-Jack polynomials are orthogonal with respect to a bilinear form of the form (p,q)↦(Lpq)(0), with Lp quantum integrals of the deformed rational Calogero–Moser–Sutherland system. In addition, we provide a new proof of the Lassalle–Nekrasov correspondence between deformed trigonometric and rational harmonic Calogero–Moser–Sutherland systems and infer orthogonality of super-Hermite polynomials, which provide joint eigenfunctions of the latter system.