Solutions of the Riccati equation and their relation to the Toda lattice

This is the authors' abstract. We don't add key points for this paper.

Continued fraction solutions to the Riccati equation are constructed using the idea of 'form invariance' of this equation under linear fractional transformation of the dependent variable. This method has recently been applied to the scalar case and a modification and extension to the corresponding matrix equation is described. It is shown that the technique is related to Euler's method for generating continued fraction solutions to the corresponding second-order linear differential equation. Several particular examples are considered and lead to some well known continued fraction representations of standard functions. It is demonstrated that the elements of the continued fraction solution are solutions of the 'Toda lattice' equations. The authors have therefore discovered an intriguing connection between the Riccati and the above equations which deserves further investigation.