Elliptic and Hyperelliptic Solutions of Discrete Painlevé I and Its Extensions to Higher Order Difference Equations

where z and a are some parameters. We give solutions of the equation (1-1) in terms of the elliptic and hyperelliptic ψ functions. In section 2, we give an elliptic solution of this equation. Section 3 contains our main subject, which is based upon the recent studies on ψ function [C, Ma2, MÔ, Ô1, Ô2]. The ψ function is defended over an algebraic curve itself embedded in its Jacobian rather than over the Jacobian variety. Although they need slightly corrections, Cantor essentially gave a determinant expression of ψ-function and a recursion equation on the ψ-functions of genus two [C]. On the other hand, Ônishi gave another determinant expression of ψ-function [Ô2]. Recently both expressions are connected by us [MÔ]. In §3, we show that the recursion relation with a correction become (1-1) and a natural third order difference equation for a certain point in the related algebraic curve,