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The symmetric fourth Painlevé hierarchy and associated special polynomials. (English) Zbl 1202.34155
Generalizing the previous study on the fourth Painlevé equation by the second author [J. Math. Phys. 44, No. 11, 5350–5374 (2003; Zbl 1063.33029)], the authors study two families of rational solutions and associated special polynomials, called symmetric Okamoto polynomials and symmetric Hermite polynomials, for the equations in the symmetric fourth Painlevé hierarchy with affine Weyl group symmetry of type \(A_{2n}^{(1)}\). Explicit representations of these special polynomials in terms of Schur functions are given and further it is numerically shown that the distribution of the roots of these polynomials is highly regular in the complex plane.

MSC:
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
33E17 Painlevé-type functions
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