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Orthogonal Polynomials with a Singularly Perturbed Airy Weight

Chao Min, Yuan Cheng

Abstract

We study the monic orthogonal polynomials with respect to a singularly perturbed Airy weight. By using Chen and Ismail's ladder operator approach, we derive a discrete system satisfied by the recurrence coefficients for the orthogonal polynomials. We find that the orthogonal polynomials satisfy a second-order linear ordinary differential equation, whose coefficients are all expressed in terms of the recurrence coefficients. By considering the time evolution, we obtain a system of differential-difference equations satisfied by the recurrence coefficients. Finally, we study the asymptotics of the recurrence coefficients when the degrees of the orthogonal polynomials tend to infinity.

Orthogonal Polynomials with a Singularly Perturbed Airy Weight

Abstract

We study the monic orthogonal polynomials with respect to a singularly perturbed Airy weight. By using Chen and Ismail's ladder operator approach, we derive a discrete system satisfied by the recurrence coefficients for the orthogonal polynomials. We find that the orthogonal polynomials satisfy a second-order linear ordinary differential equation, whose coefficients are all expressed in terms of the recurrence coefficients. By considering the time evolution, we obtain a system of differential-difference equations satisfied by the recurrence coefficients. Finally, we study the asymptotics of the recurrence coefficients when the degrees of the orthogonal polynomials tend to infinity.
Paper Structure (5 sections, 6 theorems, 85 equations)

This paper contains 5 sections, 6 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Ladder operators and the associated compatibility conditions
  3. The $t$ evolution and differential-difference equations
  4. Large $n$ asymptotics of the recurrence coefficients
  5. Conclusions

Key Result

Lemma 2.1

We have where $R_{n},\; r_n$ and $R_{n}^*,\; r_{n}^*$ are the auxiliary quantities defined by and

Theorems & Definitions (12)

  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • ...and 2 more