Bilinear Structure and Exact Solutions of the Discrete Painlev\'e I Equation

Bilinear structure for the discrete Painlev\'e I equation is investigated. The solution on semi-infinite lattice is given in terms of the Casorati determinant of discrete Airy function. Based on this fact, the discrete Painlev\'e I equation is naturally extended to a discrete coupled system. Corresponding matrix model is also mentioned.