= Repeated characteristic root of a linear recurrence
{title2=$n^j\mu^n,\quad 0\le j<m$}
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> solutions
$$
n^j\mu^n,\qquad0\leq j<m.
$$
Substitute $y_n=\mu^nz_n$. 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.
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