Constant-coefficient linear recurrences, their starting index and their characteristic polynomial

Let $K$ be a field, let $a=(a_n{)}_{n\ge 0}$ be a sequence in $K$, and let $d\ge 1$. A constant-coefficient linear recurrence of order $d$ starting at $N$ is an identity

$a_{n+d}+c_1{a}_{n+d-1}+\cdots +c_d{a}_n=0(n\ge N),$

where $c_1,\dots ,c_d\in K$ and $c_d\mathrm{\ne }0$. Its characteristic polynomial and reciprocal denominator are

$\chi (t)=t^d+c_1{t}^{d-1}+\cdots +c_d,Q(x)=1+c_{1}x+\cdots +c_d{x}^d\mathrm{.}$

The recurrence holds from the start when $N=0$, and it is eventual when such an $N$ exists. An order-zero recurrence starting at $N$ means $a_n=0$ for every $n\ge N$; its characteristic polynomial and reciprocal denominator are both $1$. This convention makes finitely supported sequences precisely the sequences of eventual order zero.

For a bi-infinite sequence $a:\mathbb{Z}\to K$, the same displayed identity is a recurrence of order $d$ when it holds for every $n\in \mathbb{Z}$. The condition $c_d\mathrm{\ne }0$ then lets the identity be solved both forward and backward.