= Repeated characteristic root of a linear recurrence
{title2=$n^j\mu^n,\quad 0\le j<m$}
A nonzero root $\mu$ of multiplicity $m$ supplies $m$ independent solutions $n^j\mu^n$ to a constant-coefficient <linear recurrence relation>. Substitution $y_n=\mu^nz_n$ 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.
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