Special polynomials associated with rational solutions of the fifth Painlevé equation
The six Painlevé equations \((\text{P}_{\text{I}}\)-\(\text{P}_{\text{VI}})\), were discovered by Painlevé, Gambier and their other colleagues whilest studying some second-order differential equations. The Painlevé equations can be thought of as nonlinear analogues of the classical special functions. In this paper, the author investigates the zeros of the special polynomials associated with rational and algebraic solutions of a \(\text{P}_{\text{V}}\) presented in the paper, where \(\alpha \), \(\beta\), \(\gamma\) and \(S\) are arbitrary constants. In Section 2, the author studies the special polynomials associated with rational solutions of \(\text{P}_{\text{V}}\), which occur in the generic case of \(\text{P}_{\text{V}}\) when \(\delta\neq 0\). These polynomials depend on a parameter and plots are given, which illustrate the motion of the zeros as the parameter varies. The special polynomials associated with algebraic solutions of \(\text{P}_{\text{V}}\), which occur when \(\delta= 0\) and \(\gamma\neq 0\) are given in Section 3. It is well-known that \(\text{P}_{\text{V}}\) with \(\delta= 0\) and \(Y\neq 0\) can be transformed into \(\text{P}_{\text{III}}\) and so the algebraic solutions of \(\text{P}_{\text{V}}\) are expressed in terms of the special polynomials associated with rational solutions of \(\text{P}_{\text{III}}\), which also depend on parameters. In closing, in Section 4 he discusses his results.
scientific article; zbMATH DE number 2157663
scientific article