COALESCENCE CASCADES AND SPECIAL-FUNCTION SOLUTIONS FOR THE CONTINUOUS AND DISCRETE PAINLEVE EQUATIONS

We present a systematization of the special-function type solutions of continuous and discrete Painleve equations. Our method is to start fr om P-VI and construct its special solutions from the solutions of the hypergeometric equation, and by coalescence obtain the lower Painleve equations together with their special solutions. In the discrete case we study the 'symmetric', one-component, three-point mapping forms of discrete Painleve equations starting from d-P-V.