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Orthogonal Polynomials with Singularly Perturbed Freud Weights.

Chao Min1, Liwei Wang1

  • 1School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China.

Entropy (Basel, Switzerland)
|May 27, 2023
PubMed
Summary
This summary is machine-generated.

This study analyzes orthogonal polynomials using a ladder operator method. We derived key difference and differential equations for recurrence coefficients and the polynomials themselves.

Keywords:
differential and difference equationsorthogonal polynomialsrecurrence coefficientssingularly perturbed Freud weights

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Area of Science:

  • Mathematical Physics
  • Orthogonal Polynomials
  • Special Functions

Background:

  • Orthogonal polynomials are fundamental in various fields, including approximation theory and quantum mechanics.
  • Freud weight functions are crucial in the study of orthogonal polynomials, particularly in relation to differential equations.
  • Singularly perturbed problems introduce complexities that require advanced analytical techniques.

Purpose of the Study:

  • To investigate polynomials orthogonal with respect to singularly perturbed Freud weight functions.
  • To derive difference and differential-difference equations governing the recurrence coefficients.
  • To obtain differential-difference and second-order differential equations for these orthogonal polynomials.

Main Methods:

  • Utilized Chen and Ismail's ladder operator approach.
  • Derived recurrence relations for the coefficients of the orthogonal polynomials.
  • Formulated differential-difference and second-order differential equations.

Main Results:

  • Established difference equations satisfied by the recurrence coefficients.
  • Derived differential-difference equations for both recurrence coefficients and orthogonal polynomials.
  • Expressed coefficients of the orthogonal polynomials in terms of the recurrence coefficients.

Conclusions:

  • The ladder operator method provides an effective framework for analyzing orthogonal polynomials with complex weight functions.
  • The derived equations offer new insights into the spectral properties of these polynomials.
  • This work contributes to the understanding of special functions in the context of singularly perturbed problems.