On the reduction of a class of nonhomogenous linear difference equations

In this note we will be concerned with nonhomogeneous linear difference equations of the form where a(z) and b(z) are assumed to be analytic in a neighborhood of infinity and vanish there, but we consider affine transformations , where r(z) and s(z) have convergent factorial series expansions in a certain left or right half-plane, together with translations of the independent variable and we ask for (in some sense) the simplest difference equations that a given one of the above from can be transformed into using these operations, i.e. we ask for canonical representatives from each distinct equivalence class. We will show that our difference equation is reucible to a difference equation with only 1 parameter in right half-planes and to one with only 2 parameters in left half-planes; one of these parametres is a1.