# Lectures on exceptional orthogonal polynomials and rational solutions to Painlev\'e equations

@article{GomezUllate2019LecturesOE, title={Lectures on exceptional orthogonal polynomials and rational solutions to Painlev\'e equations}, author={David G'omez-Ullate and Robert Milson}, journal={arXiv: Mathematical Physics}, year={2019} }

These are the lecture notes for a course on exceptional polynomials taught at the \textit{AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications} that took place in Douala (Cameroon) from October 5-12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past ten years. In addition, some new results are presented on the construction of rational solutions to Painleve equation PIV and its higher order…

## One Citation

### Hidden symmetries and nonlinear (super)algebras

- Mathematics, Physics
- 2020

Hidden symmetries, described by higher order in momenta integrals of motion that generate nonlinear algebras, are explored at the level of classical and quantum mechanics in a variety of physical…

## References

SHOWING 1-10 OF 76 REFERENCES

### Rational solutions of higher order Painlev\'{e} systems I

- Mathematics
- 2018

This is the first paper of a series whose aim is to reach a complete classification and an explicit representation of rational solutions to the higher order generalizations of $\textrm{PIV}$ and…

### Orthogonal Polynomials and Painlevé Equations

- Mathematics
- 2017

The Riemann-Hilbert formulation of orthogonal polynomials provides a crucial bridge between disparate areas of mathematics, allowing tools developed in the context of integrable systems to be…

### Application of the τ-Function Theory¶of Painlevé Equations to Random Matrices:¶PIV, PII and the GUE

- Mathematics
- 2001

Abstract: Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We…

### Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators

- Mathematics
- 2013

The type III Hermite Xm exceptional orthogonal polynomial family is generalized to a double-indexed one Xm1,m2?> (with m1 even and m2 odd such that m2 > m1) and the corresponding rational extensions…

### Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials

- Mathematics
- 2013

We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of…

### Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2018

We consider a class of monodromy-free \Sch operators with rational potentials constituted by generalized Hermite polynomials. These polynomials defined as Wronskians of classic Hermite polynomials…

### The fourth Painlevé equation and associated special polynomials

- Mathematics
- 2003

In this article rational solutions and associated polynomials for the fourth Painleve equation are studied. These rational solutions of the fourth Painleve equation are expressible as the logarithmic…