Relations for zeros of special polynomials associated to the Painleve equations

Abstract

A method for finding relations for the roots of polynomials is presented. Our approach allows us to get a number of relations for the zeros of the classical polynomials and for the roots of special polynomials associated with rational solutions of the Painleve equations. We apply the method to obtain the relations for the zeros of several polynomials. They are: the Laguerre polynomials, the Yablonskii - Vorob'ev polynomials, the Umemura polynomials, the Ohyama polynomials, the generalized Okamoto polynomials, and the generalized Hermite polynomials. All the relations found can be considered as analogues of generalized Stieltjes relations.

arXiv comment17 pages, 5 figures

References in corpus (5)

• Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach
• Non-autonomous Hénon-Heiles Systems
• On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy
• The Yablonskii - Vorob'ev polynomials for the second Painlev'e hierarchy
• Power expansions for solution of the fourth-order analog to the first Painlevé equation

Cited by in corpus (2)

• Electrostatic partners and zeros of orthogonal and multiple orthogonal polynomials
• Irrationality of the Roots of the Yablonskii-Vorob'ev Polynomials and Relations between Them