Zeros of Stieltjes and Van Vleck polynomials and applications
type: Item
instance of: scholarly article
title: Zeros of Stieltjes and Van Vleck polynomials and applications
zbMATH Open document ID: 0589.30004
DOI: 10.1016/0022-247X(85)90297-5
author: Abdallah M. Al-Rashed
author: N. Zaheer
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: The prime interest is in locating the zeros of the system of polynomials that arise in the study of the polynomial solutions of the generalized Lamé's differential equation (GLDE) \[ \{D^ 2_ z+(\sum^{p}_{j=1}\alpha_ j/(z-a_ j))D_ z+\Phi (z)/\prod^{p}_{j=1}(z-\quad a_ j)\}w(z)=0 \] where \(\Phi\) (z) is a polynomial of degree at most p-2 (p\(\geq 2)\) and \(\alpha_ j\) and \(a_ j\) are complex constants. It is known that there are at most \(C(n+p-2,p- 2)\) Van Vleck polynomials \(V(z):=\Phi (z)\) such that the GLDE has a Stieltjes polynomial solution of degree n. The main theorem develops the notion of reflector curves and sets to locate the zeros of these polynomials relative to a prescribed location of the constants. Several classical theorems follow as special cases. Applications to problems arising in physics, fluid mechanics and the location of the complex zeros of Jacobi polynomials are discussed.
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 34A05
zbMATH DE Number: 3946522
zbMATH Keywords: Lamé's differential equation
zbMATH Keywords: Van Vleck polynomials
zbMATH Keywords: Stieltjes polynomial
zbMATH Keywords: zeros
zbMATH Keywords: Jacobi polynomials
reviewed by: Peter A. McCoy
MaRDI profile type: Publication
cites work: Zeros of Stieltjes and Van Vleck Polynomials
cites work: Q5823489
cites work: Q5531193
cites work: Q5549713
cites work: Q5667036
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description: scientific article; zbMATH DE number 3946522