Hankel Determinant Structure of the Rational Solutions for Fifth PainlevÃ© Equation

Qusay Soad Abdul-Aziz, Al-Zamil (2013) Hankel Determinant Structure of the Rational Solutions for Fifth PainlevÃ© Equation. APPLIED MATHEMATICAL SCIENCES, Vol. 7 (No. 9). pp. 445-452. ISSN 1312-885X (print) ISSN 1314-7552 (online)

Abstract

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, We compute the generating function of the entries in terms of logarithmic derivative of the Heun Confluent Function.

Item Type: Article Painlev\'{e} equations, Hankel determinant structure, Umemura polynomials. MSC 2010, the AMS's Mathematics Subject Classification > 34 Ordinary differential equationsMSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations Mr. Qusay Soad Abdul-Aziz Al-Zamil 03 Feb 2014 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/1256