On the Umemura Polynomials for the Painlevé III equation

The role of Painlevé equations in nonlinear world is well recognized. Originally Painlevé derived these equations to find new transcendental functions, and irreducibility of the solutions has been established in general by Umemura [1]. However, classical solutions of the Painlevé equations have attractive and mysterious properties. Recently, such properties of rational solutions have been studied extensively. Vorob’ev and Yablonskii have shown that the rational solutions of PII are expressed by log derivative of some polynomials which are now called the Yablonskii-Vorob’ev polynomials [2]. Okamoto have shown that the rational solutions of PIV have the same property. Moreover, he also noticed that they are located on special points in the parameter space from the point of symmetry [3]. Namely, they are in the barycenter of Weyl chamber associated with the Affine Weyl group of A (1) 2 type which is the transformation group of PIV. Umemura has shown that there exist special polynomials for PIII, PV and PVI, which admit the similar properties to those of PII and PIV [4]. Those polynomials are called the Umemura polynomials. Moreover, it is reported that they have mysterious combinatorial property [5].