Complex Exceptional Orthogonal Polynomials and Quasi-invariance

• Published in: Letters in mathematical physics 2016-05, Vol.106 (5), p.583-606
• Main Authors: Haese-Hill, William A., Hallnäs, Martin A., Veselov, Alexander P.
• Format: Article
• Language: English
• Subjects: Complex Systems; Geometry; Group Theory and Generalizations; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical
• ISSN: 0377-9017; 1573-0530
• DOI: 10.1007/s11005-016-0828-8

Consider the Wronskians of the classical Hermite polynomials H λ , l ( x ) : = Wr ( H l ( x ) , H k 1 ( x ) , … , H k n ( x ) ) , l ∈ Z ≥ 0 \ { k 1 , … , k n } , where k i = λ i + n - i , i = 1 , … , n and λ = ( λ 1 , … , λ n ) 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.