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Double Scaling Limit for Modified Jacobi-Angelesco Polynomials

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Notions of Positivity and the Geometry of Polynomials

Part of the book series: Trends in Mathematics ((TM))

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 a 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.

Mathematics Subject Classification (2000).30E15, 41A60, 42C05.

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Authors and Affiliations

  1. Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001, Leuven, Belgium

    Klaas Deschout & Arno B. J. Kuijlaars

Authors
  1. Klaas Deschout
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  2. Arno B. J. Kuijlaars
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Correspondence to Klaas Deschout .

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Editors and Affiliations

  1. , Department of Mathematics, Royal Institute of Technology, Stockholm, 100 44, Sweden

    Petter Brändén

  2. , Mathematical Department, Stockholm University, Stockholm, SE-106 91, Sweden

    Mikael Passare

  3. , Department of Mathematics, University of California, Santa Barbara, 93106-3080, California, USA

    Mihai Putinar

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Deschout, K., Kuijlaars, A.B.J. (2011). Double Scaling Limit for Modified Jacobi-Angelesco Polynomials. In: Brändén, P., Passare, M., Putinar, M. (eds) Notions of Positivity and the Geometry of Polynomials. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0348-0142-3_8

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