Elliptic and Hyperelliptic Solutions of Discrete Painlev\'e I and Its Extensions to Higher Order Difference Equations

The solutions of the discrete Painlev\'e equation I were constructed in terms of elliptic and hyperelliptic $\psi$ functions for algebraic curves of genera one and two. For the case of genus two, there appear higher order difference equations which naturally contain the discrete Painlev\'e equation I as a special case.