**Theorem 1**
Let

be a polynomial with

and

.
Factor

over the complex numbers as

where

are distinct nonzero complex numbers,
and

are positive integers.
Then a sequence

satisfies the linear recurrence
with characteristic polynomial

if and only if
there exist polynomials

,

, ...,

with

for

such that

for all $n$

**Corollary 2**
Suppose in addition that

has no repeated factors;
in other words suppose that

.
Then

where

are distinct nonzero complex numbers
(the roots of

).
Then

satisfies the linear recurrence
with characteristic polynomial

if and only if
there exist constants

(not depending on

)
such that

for all $n$