Double scaling limit for modified Jacobi-Angelesco polynomials

Details Double scaling limit for modified Jacobi-Angelesco polynomials Double scaling limit for modified Jacobi-Angelesco polynomials

2011-02-07

arxiv.org/abs/1102.1349v1

Notions of Positivity and the Geometry of Polynomials (P. Br\"and\'en, M. Passare, and M. Putinar, eds.), Trends in Mathematic

Mathematics

Classical Analysis and ODEs

47 pages, 9 figures

Scientific paper

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.

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