Asymptotics of Polynomials Orthogonal With Respect to a Generalized Freud Weight With Application to Special Function Solutions of Painlevé-IV

2023/12/18 by Ahmad Barhoumi, Barhoumi, Ahmad · 1 citation
Chemistry · Physics and Astronomy · #30E15 #33C47 #34E05 #34M50 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #Primary: 34M55 #Quantum chaos and dynamical systems #Secondary: 15B52

paper · pdf · doi:10.48550/arxiv.2312.11294

openalex publication_date 2023/12/18 · openalex created_date 2023/12/20 · openalex updated_date 2026/07/28

Abstract

We obtain asymptotics of polynomials satisfying the orthogonality relations ∫_ℝ z^k P_n(z; t , N) e^{-N ((1)/(4)z4 + (t)/(2)z2 )} d z = 0 for k = 0, 1, ..., n-1, where the complex parameter t is in the so-called two-cut region. As an application, we deduce asymptotic formulas for certain families of solutions of Painlevé-IV which are indexed by a non-negative integer and can be written in terms of parabolic cylinder functions. The proofs are based on the characterization of orthogonal polynomials in terms of a Riemann-Hilbert problem and the Deift-Zhou non-linear steepest descent method.

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