(Multiple roots of Characteristic Polynomial.) Let
$\rho(\xi) = \sum_{j=0}^{k} \alpha_j \xi^j$
(1)
and consider the Difference Equation
$\sum_{j=0}^{k} \alpha_j y_{n+j} = 0.$
(2)
Suppose that $r$ is a root of multiplicity $q$, i.e.,
$\rho(r) = \rho'(r) = \dots = \rho^{(q-1)}(r) = 0.$
(3)
Show that the sequence
$y_n = n^s r^n$
(4)
is a solution of the difference equation for all $s = 0, 1, \dots, q - 1.$
