Zero coefficient of associated homogeneous recurrence relation

Question

When solving a non homogeneous recurrence relation, is it possible for a coefficient in the associated homogeneous equation to be zero? Meaning the solution might consist solely of the particular solution?

Answer 1

Yes, the general solution of a linear recurrence has the form $\alpha_1 h_1(n) + \ldots \alpha_r h_r(n) + p(n)$, where the $h_i$ are the linearly independent solutions to the homogeneous equation, $p$ is a particular solution, and the $\alpha_i$ are arbitrary constants. It is certainly possible to have $\alpha_1 = \alpha_2 = \ldots = \alpha_r = 0$.

In any case, note that $p$ is a particular solution, you might as well take the solution to your problem as $p$, in which case all the $\alpha_i = 0$.