Special function solutions of the discrete painlevé equations

In this paper, we present a review of the special solutions of discrete (difference) and q-discrete Painlevé equations in terms of discrete special functions. These solutions exist whenever the parameters of the Painlevé equation satisfy a particular constraint. The discrete special functions belong to the hypergeometric family, although in some cases they seem to go beyond the known discrete and q-discrete hypergeometric functions. The equations studied in this paper are chosen on the basis of our recent classification of discrete Painlevé equation with the help of affine Weyl groups.