Analytic solutions of the difference equation Psi(z + h) = M(z)Psi(z) are explored. Here z is a complex variable, h > 0 is a parameter, and M is a given SL(2, C)-valued function. It is assumed that M either is analytic in a bounded domain or is a trigonometric polynomial. A simple method to derive the asymptotics of solutions as h -> 0 is described.