 Exponential forcing in a linear recurrence

Let $E$ be the forward shift on sequences and $P$ the polynomial defining a constant-coefficient linear recurrence relation. For a nonzero base $r$, $P(E)r^n=P(r)r^n$, so $r^n/P(r)$ is a particular solution when $P(r)\ne 0$. At a simple root of the characteristic equation of a linear recurrence, the correct trial acquires a factor $n$:The identity follows by substituting $(n+j)r^{n+j}$ into each shift term. A root of multiplicity $m$ analogously requires a degree-$m$ polynomial factor; the lower-degree homogeneous terms are supplied by the repeated characteristic root of a linear recurrence.