General Solution of the Three-site Master Equation and the Discrete Riccati Equation

We first obtain by analogy with the continuous (differential) case the general solution of the discrete Riccati equation. Moreover, we establish the full equivalence between our discrete Riccati equation and a corresponding homogeneous second order discrete linear equation. Our results can be considered the discrete analog of Mielnik's construction in supersymmetric quantum mechanics [J. Math. Phys. 25, 3387 (1984)]. We present an application to the three-site master equation obtaining explicitly the general solutions for the simple cases of free random walk and the biased random walk. We consider the continuous Riccati equation (CRE) y ′ = a(x)y 2 + b(x)y + c(x) with the known particular solution y 0 and let y 1 = u + y 0 be the second solution. By substituting y 1 in CRE one gets the Bernoulli equation u ′ = au 2 + (2ay 0 + b)u. Furthermore, using v = 1/u, the simple first-order linear differential equation v ′ + (2ay 0 + b)v + a = 0 is obtained, which can be solved by employing the integration