Repeated characteristic root of a linear recurrence ($n^j{\mu }^n,0\le j<m$)

A nonzero root $\mu$ of multiplicity $m$ supplies $m$ independent solutions $n^j{\mu }^n$ to a constant-coefficient linear recurrence relation. Substitution $y_n={\mu }^n{z}_n$ reduces its repeated factor to the $m$th forward difference operator. A zero root needs separate initial-data treatment: for characteristic polynomial ${\lambda }^m$, the recurrence is $y_{n+m}=0$, leaving the first $m$ values arbitrary.