Zeroes of a Recurrence Relation www.carom-maths.co.uk Activity 2-17: Zeroes of a Recurrence Relation

We all know the Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13... We can write this as un+1= un + un-1, with u1 = 1, u2 = 1. So the next term is the sum of the previous two, and we have some initial conditions. We have here an example of a Linear Recurrence Relation (or LRS).

un+1 = anun + an-1un-1 + an-2un-2 +......+ an-k+1un-k+1 is the general LRS of order k. So the next term is a linear combination of the last k terms (we also need initial conditions). We will only be looking at the cases where every ai is an integer: we will call these ‘integer LRSs’. Now there is a way to calculate future terms for an LRS using a matrix. Let’s demonstrate this using the Fibonacci sequence.

and if we have some computer help in taking the powers of a matrix, So it is clear that and if we have some computer help in taking the powers of a matrix, we have a good way of calculating future terms of an LRS.

Task: use this spreadsheet to find the 21st term, that’s F21, in the Fibonacci sequence. Taking Powers of a Matrix spreadsheet The 21st term in the Fibonacci sequence is 10 946.

Task: given the LRS un+1= un + un-1 + un-2 with u1 = 1, u2 = 1, u3 = 1, find the 22nd term. Using a 3 x 3 matrix this time: The 22nd term in this order-3 LRS sequence is 157 305.

Now we can note that it makes perfect sense to run the Fibonacci sequence backwards. ...5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13... So u0 = 0, u-1 = 1, u-2 = -1, and so on: the rule still holds. Now a further question: how many 0s are there in the Fibonacci sequence? It seems clear that there can only be one. In that case, how many 0s can any order-2 LRS have? Can it have an infinite number? Or is there a maximum?

It is easy to construct an LRS that does have an infinite number of 0s. Consider un+2 = 2un, with u1 = 0, u2 = 2. This gives us the sequence 0, 2, 0, 4, 0, 8, 0, 16... This clearly has an infinite number of 0s, BUT we call this type of LRS degenerate. The LRS un+1 = anun + an-1un-1 has associated with it the characteristic equation 2 = an + an-1. This has two roots, 1 and 2, and if their ratio is a root of unity, we say the LRS is degenerate.

So for our example un+2 = 2un, the characteristic equation is 2 = 2, and so 1, 2 are ±√2, and their ratio is -1. The solution to an LRS is un = A1n+ B2n . We can find A and B from the initial conditions. So for our Fibonacci sequence, the characteristic equation is 2  1 = 0, which gives solutions 1, 2 = Using u1 = 1, u2 = 1, we find A = , B = , and so Task: test this out for various n. Fn =

So let’s go back to our question and rephrase it: how many 0s can a non-degenerate order-2 LRS have? Theorem: Skolem-Mahler-Lech (proved in 1953). Any non-degenerate integer LRS has a finite number of 0s. It can also be proved that the largest number of 0s an order-2 non-degenerate integer LRS can have is 1, (so the Fibonacci sequence has the maximum). How can we prove this?

The order-2 LRS un+1 = anun + an-1un-1 has associated with it the characteristic equation 2 = an + an-1. Suppose the roots are 1, 2 . So we have un = A1n+ B2n . Suppose un = 0 for n = p and n = q, with p > q. 0 = A1p+ B2p  Similarly, Subtracting, we have , and thus the ratio of the roots is a root of unity, and the LRS is degenerate. So the largest number of 0s an order-2 non-degenerate integer LRS can have is 1.

What is the largest number of 0s an order-3 non-degenerate integer LRS can have? Beukers has proved (1991) that the answer is 6. Is it possible to find an order-3 LRS with six 0s? The mathematician Berstel managed to do exactly that.

Task: using our matrix spreadsheet, see if you can find all six zeroes there are to be found. So we have zeroes for a0, a1, a4, a6, a13, and rather surprisingly, for a52.

With thanks to: Graham Everest and Tom Ward. Carom is written by Jonny Griffiths, mail@jonny-griffiths.net