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Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type E8(1)

We present special solutions of the discrete Painleve equations associated with A0(1), A0(1)* and A0(1)**-surfaces. These solutions can be expressed by solutions of linear difference equations. Here the A0(1)-surface discrete Painleve equation is the most generic difference equation, as all discrete Painleve equations can be obtained by its degeneration limit. These special solutions exist when the parameters of the discrete Painleve equation satisfy a particular constraint. We consider that these special functions belong to the hypergeometric family although they seem to go beyond the known discrete and q-discrete hypergeometric functions. We also discuss the degeneration scheme of these solutions.