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

For a constant-coefficient linear recurrence relation over the complex numbers, a nonzero root $\mu$ of multiplicity $m$ in its characteristic polynomial of a linear recurrence supplies the $m$ linearly independent solutionsSubstitute $y_n={\mu }^n{z}_n$. Each factor $E-\mu$ of the shift linear operator becomes $\mu$ times the forward difference operator on $z_n$, so its $m$th power kills every polynomial in $n$ of degree less than $m$. Their independence follows from the Lagrange root bound over a field. The same argument works over a field of characteristic zero; the displayed monomials need modification in positive characteristic.

A zero root needs separate initial-data treatment: for characteristic polynomial ${\lambda }^m$, the recurrence relation is $y_{n+m}=0$, leaving the first $m$ values arbitrary.