Is a constant sequence a recurrence relation? What is its order?

Question

My textbook defines a recurrence relation as a sequence whose nth term can be expressed as a function of the previous terms in the sequence.

Would it be incorrect to say that $a_n = 5$ is a recurrence relation? I could pose the argument that it is a function of the last in element, in the sense that the function is just $f(n) = 5$. If it is a recurrence relation, is it valid to call it a zeroth order recurrence relation?

Answer 1

To stay with your example:

$a_n = 5\; (n \in \mathbb{N)}$ is not a recurrence relation but an explicit formula defining a polynomial sequence of order 0, because the defining expression for the sequence is a polynomial of degree 0.

But it can also be written as a recurrence relation:

$a_n = a_{n-1}, a_1 = 5$.

Answer 2

If you want it to be a homogeneous linear recurrence, then it has order 1 ($a_n=a_{n-1}$, characteristic polynomial $x-1$).

Otherwise I'm not sure what you mean by "order". It could certainly be "zeroth order" in some sense of the word.

Answer 3

Your book is wrong. (What book is that?) A recurrence relation is not a sequence of numbers. A solution of a recurrence relation is a sequence of numbers.

The constant sequence $a_n=5$ is the solution of lots of different recurrence relations. For instance, it is a solution of the second order recurrence relation $$a_n=2a_{n-1}-a_{n-2}.$$ The general solution of that recurrence is $$a_n=C_1+C_2n$$ where $C_1$ and $C_2$ are arbitrary constants.