On quasi-orthogonal polynomials: Their differential equations, discriminants and electrostatics

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:474

Article

Ismail, Mourad E. H.¹  Wang, Xiang-Sheng² 

[1] Univ Cent Florida, Dept Math, Orlando, FL 32816 USA

[2] Univ Louisiana Lafayette, Dept Math, Lafayette, LA 70503 USA

关键词: Quasi-orthogonal polynomials;    Discriminant;    Electrostatics;    Hamburger moment problem;    Nonlinear orthogonality preserving transformation;   

DOI  :  10.1016/j.jmaa.2019.02.011

来源: Elsevier

【 摘 要 】

In this paper, we develop a general theory of quasi-orthogonal polynomials. We first derive three-term recurrence relation and second-order differential equations for quasi-orthogonal polynomials. We also give an expression for their discriminants in terms of the recursion coefficients of the corresponding orthogonal polynomials. In addition, we investigate an electrostatic equilibrium problem where the equilibrium position of movable charges is attained at the zeros of the;quasi-orthogonal polynomials. The examples of the Freud weight w(x) = e(-x4+2tx2) and the Jacobi weight w(x) = (1 - x)(alpha) (1 + x)(beta) are discussed in some detail. Finally, we consider the nonlinear orthogonality preserving transformation and related matrix problem. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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