A Determinant Formula for a Class of Rational Solutions of Painlevé V Equation

It is known that six Painlevé equations are in general irreducible, namely, their solutions cannot be expressed by “classical functions” in the sense of Umemura [18]. However, it is also known that they admit classical solutions for special values of parameters except for PI. Much effort have been made for the investigation of classical solutions. As a result, it has been recognized that there are two classes of classical solutions. One is transcendental classical solutions expressible in terms of functions of hypergeometric type. Another one is algebraic or rational solutions. It is also known that the Painlevé equations (except for PI) admit action of the affine Weyl groups as groups of the Bäcklund transformations. It is remarkable that such classical solutions are located on special places from a point of view of symmetry in the parameter spaces [12, 13, 14, 15]. For example, PII, PIII and PIV, whose symmetry is described by the affine Weyl group of type A (1) 1 , A (1) 1 ×A (1) 1 and A (1) 2 , respectively, admit transcendental classical solutions on the reflection hyperplanes, and rational solutions on the barycenters of Weyl chambers of the corresponding affine Weyl group. Umemura et al have investigated the class of solutions on the barycenters of Weyl chambers and found that (1) these solutions are expressed by some characteristic polynomials generated by the Toda type bilinear equations, (2) the coefficients of such polynomials admit mysterious combinatorial properties [17, 7, 16]. These special polynomials are sometimes referred as Yablonskii-Vorob’ev polynomials for PII [19], Okamoto polynomials for PIV [14], Umemura polynomials for PIII, PV and PVI. One important aspect among such polynomials is that they are expressed as special cases of the Schur functions. As is well known, the Schur functions are characters of the irreducible polynomial representations of GL(n) and arise as τ -functions of the KP hierarchy [1]. For example, it is known that the special polynomials for PII and PIII are expressible by 2-reduced Schur functions, and those for PIV by 3-reduced Schur functions [2, 3, 4, 8]. In this paper, we consider PV,