Coefficients of Wronskian Hermite polynomials

Abstract

We study Wronskians of Hermite polynomials labelled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behaviour of the Wronskian Hermite polynomials when the length of the core tends to infinity, while fixing the quotient. Via this combinatorial setting, we obtain in a natural way the generalization of the correspondence between Hermite and Laguerre polynomials to Wronskian Hermite polynomials and Wronskians involving Laguerre polynomials. Lastly, we generalize most of our results to polynomials that have zeros on the $p$-star.

arXiv comment41 pages

References in corpus (17)

• An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
• Determinant Structure of the Rational Solutions for the Painlevé IV Equation
• Extended Krein-Adler theorem for the translationally shape invariant potentials
• Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators
• Exceptional Jacobi polynomials
• On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy
• Exceptional Laguerre polynomials
• Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger's equation
• Poles of Painlevé IV Rationals and their Distribution
• Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials
• Complex exceptional orthogonal polynomials and quasi-invariance
• Exceptional orthogonal polynomials and generalized Schur polynomials
• Recurrence Relations for Wronskian Hermite Polynomials
• Wronskian Appell Polynomials and Symmetric Functions

Cited by in corpus (7)

• Pattern transformation in higher-order lumps of the Kadomtsev-Petviashvili I equation
• Classification of KPI Lumps
• Counting monster potentials
• Rational Solutions of the Fifth Painlevé Equation. Generalised Laguerre Polynomials
• The Adelic Grassmannian and Exceptional Hermite Polynomials
• Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence
• Asymptotic behavior of Wronskian polynomials that are factorized via $p$-cores and $p$-quotients