# Asymptotics for Recurrence Coefficients of X1-Jacobi Polynomials and Christoffel Function

## TL;DR

This paper derives asymptotic behaviors of recurrence coefficients for X1-Jacobi polynomials and explores their impact on the Christoffel function, addressing challenges posed by their non-standard recurrence relations.

## Contribution

It provides new asymptotic formulas for recurrence coefficients of X1-Jacobi polynomials and analyzes their influence on Christoffel functions, overcoming the limitations of classical methods.

## Key findings

- Asymptotic formulas for recurrence coefficients derived
- Limit behavior of Christoffel function analyzed
- Handling of five-term recurrence relations in exceptional polynomials

## Abstract

Computing asymptotics of the recurrence coefficients of X1-Jacobi polynomials we investigate the limit of Christoffel function. We also study the relation between the normalized counting measure based on the zeros of the modified average characteristic polynomial and the Christoffel function in limit. The proofs of corresponding theorems with respect to ordinary orthogonal polynomials are based on the three-term recurrence relation. The main point is that exceptional orthogonal polynomials possess at least five-term formulae and so the Christoffel-Darboux formula also fails. It seems that these difficulties can be handled in combinatorial way.