  We investigate and derive second solutions to linear homogeneous second-order
difference equations using a variety of methods, in each case going beyond the
purely formal solution and giving explicit expressions for the second solution.
We present a new implementation of d'Alembert's reduction of order method,
applying it to linear second-order recursion equations. Further, we introduce
an iterative method to obtain a general solution, giving two linearly
independent polynomial solutions to the recurrence relation. In the case of a
particular confluent hypergeometric function for which the standard second
solution is not independent of the first, i.e. the solutions are degenerate, we
use the corresponding differential equation and apply the extended
Cauchy-integral method to find a polynomial second solution for the difference
equation. We show that the standard d'Alembert method also generates this
polynomial solution.
