Irrationality of the Roots of the Yablonskii-Vorob'ev Polynomials and Relations between Them

TL;DR

This paper investigates the roots of Yablonskii-Vorob'ev polynomials, proving their irrationality and uncovering relations between roots of consecutive degrees through analysis of rational solutions to the second Painlevé equation.

Contribution

It establishes the irrationality of the roots and derives new relations between roots of consecutive polynomials, advancing understanding of these special polynomials.

Findings

Nonzero roots of Yablonskii-Vorob'ev polynomials are irrational.

Divisibility properties of polynomial coefficients related to powers of 4.

Relations between roots of consecutive degree polynomials.

Abstract

We study the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlev\'e equation. Divisibility properties of the coefficients of these polynomials, concerning powers of 4, are obtained and we prove that the nonzero roots of the Yablonskii-Vorob'ev polynomials are irrational. Furthermore, relations between the roots of these polynomials for consecutive degree are found by considering power series expansions of rational solutions of the second Painlev\'e equation.