Special function solutions of the discrete Painleve equations

In this paper, we present a review of the special solutions of discrete (di fference) and q-discrete Painleve equations in terms of discrete special fu nctions. These solutions exist whenever the parameters of the Painleve equa tion satisfy a particular constraint. The discrete special functions belong to the hypergeometric family, although in some cases they seem to go beyon d 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 Painleve equations with the help of affine Weyl groups. (C) 20 01 Elsevier Science Ltd. All rights reserved.