Special polynomials associated with the Painlevé equations I

The Painlevé equations have rational or algebraic solutions on special parameters. We can find rational or algebraic solutions of the Painlevé equations as fixed points of the Bäcklund transformations. The τ function of the rational or algebraic solution can be written as the product of a special polynomial and an exponential factor. Since a series of τ functions satisfies the Toda equation, we obtain a recursive relation of the special polynomials. The coefficients of the special polynomials for the sixth Painlevé equation are described by the Young diagram. (The original manuscript by the author was submitted to the proceeding of the Montreal conference in 1996, which were not published. The abstract was not part of the original manuscript and has not been written by the author.)