# Wronskian Appell Polynomials and Symmetric Functions

## TL;DR

This paper explores Wronskian Appell polynomials linked to symmetric functions, providing new formulas, recurrence relations, and proving an integrality conjecture, with applications in Painlevé equations and orthogonal polynomials.

## Contribution

It introduces new properties, recurrence relations, and proves an integrality conjecture for Wronskian Hermite polynomials, connecting them to symmetric function theory.

## Key findings

- Derived formulas for derivatives, averages, and variances of Wronskian Appell polynomials
- Established recurrence relations for these polynomials
- Proved an integrality conjecture for Wronskian Hermite polynomials

## Abstract

We study Wronskians of Appell polynomials indexed by integer partitions. These families of polynomials appear in rational solutions of certain Painlev\'e 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.