Double scaling limit for modified Jacobi-Angelesco polynomials

TL;DR

This paper investigates the asymptotic behavior of modified Jacobi-Angelesco polynomials near a critical transition point using Riemann-Hilbert analysis and introduces a new local parametrix for the double scaling limit.

Contribution

It provides a Mehler-Heine type asymptotic formula for these polynomials in a double scaling regime, advancing understanding of their asymptotic properties at a critical transition.

Findings

Derived a Mehler-Heine asymptotic formula in the double scaling limit.

Developed a new local parametrix for Riemann-Hilbert analysis.

Analyzed the transition at a = -1 for modified Jacobi weights.

Abstract

We consider multiple orthogonal polynomials with respect to two modified Jacobi weights on touching intervals [a,0] and [0,1], with a < 0, and study a transition that occurs at a = -1. The transition is studied in a double scaling limit, where we let the degree n of the polynomial tend to infinity while the parameter a tends to -1 at a rate of O(n^{-1/2}). We obtain a Mehler-Heine type asymptotic formula for the polynomials in this regime. The method used to analyze the problem is the steepest descent technique for Riemann-Hilbert problems. A key point in the analysis is the construction of a new local parametrix.