Wronskian Appell Polynomials and Symmetric Functions

Abstract

We study Wronskians of Appell polynomials indexed by integer partitions. These families of polynomials appear in rational solutions of certain Painlevé equations and in the study of exceptional orthogonal polynomials. We determine their derivatives, their average and variance with respect to Plancherel measure, and introduce several recurrence relations. In addition, we prove an integrality conjecture for Wronskian Hermite polynomials previously made by the first and last authors. Our proofs all exploit strong connections with the theory of symmetric functions.

18 pages

References in corpus (9)

• Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
• Determinant Structure of the Rational Solutions for the Painlevé II Equation
• On the Umemura Polynomials for the Painlevé III equation
• Rational Solutions of the Painleve' VI Equation
• Exceptional Jacobi polynomials
• Exceptional Laguerre polynomials
• A unified matrix approach to the representation of Appell polynomials
• Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials
• Exceptional orthogonal polynomials and generalized Schur polynomials

Cited by in corpus (4)

• Coefficients of Wronskian Hermite polynomials
• Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence
• Asymptotic behavior of Wronskian polynomials that are factorized via $p$-cores and $p$-quotients
• Recurrence relations for Wronskian Laguerre polynomials