Riccati difference equation

Summary

The Riccati difference equation is the first-order rational difference equation $x_{n+1} = (a x_n + b)/(c x_n + d)$ with $c \ne 0$. It is the iteration of a Möbius transformation. A substitution turns it into a linear recurrence of order two, so it can be solved in closed form. It is the discrete analogue of the Riccati equation for differential equations, which is linearized by a similar substitution.