Hankel Determinant structure of the Rational Solutions for Painlevé Five

Details Hankel Determinant structure of the Rational Solutions for Painlevé Five Hankel Determinant structure of the Rational Solutions for Painlevé Five

2009-04-03

arxiv.org/abs/0904.0640v1

Mathematics

Analysis of PDEs

7 Pages

Scientific paper

In this paper, we construct the Hankel determinant representation of the rational solutions for the fifth Painlev\'{e} equation through the Umemura polynomials. Our construction gives an explicit form of the Umemura polynomials $\sigma_{n}$ for $ n\geq 0$ in terms of the Hankel Determinant formula. Besides, this result also gives another proof of the fact that the $\sigma_{n}$'s are indeed polynomials. We compute the generating function of the entries in terms of logarithmic derivative of the Heun Confluent Function.

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