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Complex exceptional orthogonal polynomials and quasi-invariance

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posted on 2016-02-18, 12:46 authored by William Haese-Hill, Alexander VeselovAlexander Veselov, Martin Hallnas
Consider the Wronskians of the classical Hermite polynomials Hλ₁(x):= Wr(Hl(x);Hk1 (x)…;Hkn(x)); l ϵ Z≥0 \{k1; : : : ; kn}; where ki = λ₁ + n - i; i = 1;…, n and λ = (λ₁;…; λn) is a partition. Gómez-Ullate et al. showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with respect to a certain inner product, but in contrast to the usual case have some degrees missing (so called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of C[x] satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schrödinger operators, is also presented.

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School

  • Science

Department

  • Mathematical Sciences

Published in

Letters in Mathematical Physics

Citation

HAESE-HILL, W., VESELOV, A.P. and HALLNAS, M., 2016. Complex exceptional orthogonal polynomials and quasi-invariance. Letters in Mathematical Physics, 106(5), pp.583-606.

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© Springer Verlag (Germany)

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  • VoR (Version of Record)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by/4.0/

Publication date

2016

Notes

This is an Open Access Article. It is published by Springer under the Creative Commons Attribution 4.0 Unported Licence (CC BY). Full details of this licence are available at: http://creativecommons.org/licenses/by/4.0/

ISSN

1573-0530

Language

  • en

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