The Fourth‐Order Discrete Painlevé I Equation Under Nonzero Boundary Conditions

ABSTRACT Under the zero boundary conditions, a new generalized nonisospectral Volterra lattice as well as its Lax pair and solutions are established based on the nonisospectral deformation of the symmetric orthogonal polynomials. As their stationary reductions, the corresponding fourth‐order discrete Painlevé I equation is derived together with its Lax pair and solutions. Furthermore, the above results are successfully extended to the nonzero boundary conditions. Lax pair and solutions are derived for the generalized nonisospectral Volterra lattice under the nonzero boundary conditions. Correspondingly, the stationary reduction method is applied to obtain the fourth‐order discrete Painlevé I equation under nonzero boundary conditions for the first time.