Masuda, Tetsu On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. (English) Zbl 1151.34340 Funkc. Ekvacioj, Ser. Int. 46, No. 1, 121-171 (2003). The author considers a certain class of algebraic solutions of the sixth Painlevé equation \(P_{VI}\) (in Hamiltonian form), for which he presents a determinant formula. The entries of the determinant are essentially the Jacobi polynomials. The well known fact that each of the Painlevé equations can be obtained from \(P_{VI}\) by a coalescence procedure is then used to obtain, from this family of algebraic solutions of \(P_{VI}\), rational solutions of \(P_{V}\), \(P_{III}\) and \(P_{II}\). Finally, the author considers the connection with the Umemura polynomials for \(P_{VI}\). Reviewer: Andrew Pickering (Madrid) (MR1996296) Cited in 16 Documents MSC: 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 35F20 Nonlinear first-order PDEs PDFBibTeX XMLCite \textit{T. Masuda}, Funkc. Ekvacioj, Ser. Int. 46, No. 1, 121--171 (2003; Zbl 1151.34340) Full Text: DOI arXiv Digital Library of Mathematical Functions: §32.9(iii) Sixth Painlevé Equation ‣ §32.9 Other Elementary Solutions ‣ Properties ‣ Chapter 32 Painlevé Transcendents