General solution of a second order non-homogenous linear difference equation with noncommutative coefficients

where the unknown {Yp}p∈N as well as the non-homogenous term {φp}p∈N are sequences from a vectorial space V , and the coefficients L0,L1, are linear noncommutative operators mapping V on itself, independent from the discrete variable p ∈ N. This equation encompasses interesting problems arising in very different scenarios. If, for instance, the reference space V is the complex Euclidean space Cn, that is Yp and φp+1 are n-dimensional vectors, L0 and L1 n × n complex matrices, then eq. (1) is the vectorial representation of a system of second-order linear nonhomogenous difference equations. As another example, let’s identify V as the vectorial space of all linear operators defined on a given Hilbert space. Now, the operators L0 and L1 act upon operators and for this reason are called superoperators. The master equations appearing in the theory of open quantum systems provide examples of equations belonging to this class [5]. It is of relevance to emphasize from the very beginning that the ingredients Yp, φp, L0 and L1 of eq. (1) may be also interpreted as elements of an assigned algebra V . Let’s consider, for example, V as the