A Miura of the Painlevé I Equation and Its Discrete Analogs

We present Miura transformations for the continuous and several discrete Painlevé I equations. In the case of the continuous PI, we use the Hamiltonian formulation of the Painlevé equations and show that there exists a Miura transformation between PI and the binomial, second degree, equation of Cosgrove SDV. In the case of the discrete PI's we obtain two different kinds of Miuras. One kind relates a d-PI to some other d-PI while the other leads to discrete four-point equations which are the discrete analogs of the derivative of Cosgrove's equation SDV.