The generalized Yablonskii–Vorob'ev polynomials and their properties

Abstract

Rational solutions of the generalized second Painlevé hierarchy are classified. Representation of the rational solutions in terms of special polynomials, the generalized Yablonskii–Vorob'ev polynomials, is introduced. Differential–difference relations satisfied by the polynomials are found. Hierarchies of differential equations related to the generalized second Painlevé hierarchy are derived. One of these hierarchies is a sequence of differential equations satisfied by the generalized Yablonskii–Vorob'ev polynomials.