Abstract
We present a finite-dimensional system of discrete orthogonality relations for the Hall-Littlewood polynomials. A compact determinantal formula for the weights of the discrete orthogonality measure is formulated in terms of a Gaudin-type conjecture for the normalization constants of a dual system of orthogonality relations. The correctness of our normalization conjecture has been checked in some special cases: for Hall-Littlewood polynomials up to four variables (i), for the reduction to Schur polynomials (ii), and in a continuum limit in which the Hall-Littlewood polynomials degenerate into the Bethe Ansatz eigenfunctions of the Schrödinger operator for identical Bose particles on the circle with pairwise delta-potential interactions (iii).
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Work supported in part by the Anillo Ecuaciones Asociadas a Reticulados financed by the World Bank through the Programa Bicentenario de Ciencia y Tecnología, and by the Programa Reticulados y Ecuaciones of the Universidad de Talca.
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van Diejen, J.F. Finite-Dimensional Orthogonality Structures for Hall-Littlewood Polynomials. Acta Appl Math 99, 301–308 (2007). https://doi.org/10.1007/s10440-007-9168-0
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DOI: https://doi.org/10.1007/s10440-007-9168-0