Comparable solutions to nth-order linear difference equations

We consider the nth-order linear difference equation lny(t) = Lny(t) + q(t)y (t + [n2]) = 0, for t ϵ [a, b] where q(t) is a real-valued function defined on [a, b]. We define the (formal) adjoint operator l∗n of ln by l∗nz(t) = L∗nz(t) + (−1)nq(t)z (t + [n2]), for t ϵ [a, b]. We compare boundary value solutions of lny(t) = 0 to similar solutions of the adjoint equation l∗nz(t) = 0.