Painlevé equations and orthogonal polynomials on the unit circle

• Grant number: 22/09575-5
• Support Opportunities: Regular Research Grants
• Start date: October 01, 2022
• End date: September 30, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Cleonice Fátima Bracciali
• Grantee: Cleonice Fátima Bracciali
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Associated researchers: Luana de Lima Silva Ribeiro

Abstract

There is a very interesting relation between orthogonal polynomials on the real line and Painlevé equations. More explicitly, the coefficients of the recurrence relation of semiclassical orthogonal polynomials generally lead to solutions of Painlevé equations (differential or discrete). Solutions of these Painlevé equations in terms of classical special functions are the most relevant. Furthermore, the theory of orthogonal polynomials often provides new insights into the behaviour of solutions to Painlevé equations. Some integrable systems, such as Toda lattices, Langmuir lattices and systems of differential differences, also appear in a very natural way in the theory of classical orthogonal polynomials. It is interesting to investigate whether such relations are present in non-classical orthogonal polynomials. (AU)