The third Painlevé equation and associated special polynomials

In this paper we are concerned with rational solutions, algebraic solutions and associated special polynomials with these solutions for the third Painlevé equation (PIII). These rational and algebraic solutions of PIII are expressible in terms of special polynomials defined by second-order, bilinear differential-difference equations which are equivalent to Toda equations. The structure of the roots of these special polynomials is studied and it is shown that these have an intriguing, highly symmetric and regular structure. Using the Hamiltonian theory for PIII, it is shown that these special polynomials satisfy pure difference equations, fourth-order, bilinear differential equations as well as differential-difference equations. Further, representations of the associated rational solutions in the form of determinants through Schur functions are given.