Relations for zeros of special polynomials associated to the Painleve equations

Details Relations for zeros of special polynomials associated to the Painleve equations Relations for zeros of special polynomials associated to the Painleve equations

2006-10-24

arxiv.org/abs/nlin/0610058v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

17 pages, 5 figures

Scientific paper

10.1016/j.physleta.2007.03.081

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.

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