In The Da Vinci Code, Dan Brown feels he need to bring in a French cryptologist, Sophie Neveu, to explain the mystery behind this series of numbers:
13 – 3 – 2 – 21 – 1 – 1 – 8 – 5
The Fibonacci sequence, 1-1-2-3-5-8-13-21-34-55-89-144-… is such that any number in it is the sum of the two previous numbers.
It is the most famous of all integral linear recursive sequences, that is, a sequence of integers
such that there is a monic polynomial with integral coefficients of a certain degree
such that for every integer
For the Fibonacci series
The set of all integral linear recursive sequences, let’s call it
For starters, it is a ring. That is, we can add and multiply such sequences. If
then the sequences
are again linear recursive. The zero and unit in this ring are the constant sequences
So far, nothing terribly difficult or exciting.
It follows that
sending a sequence
It’s a bit more difficult to see that
with properties dual to those of usual multiplication.
To describe this co-multiplication in general will have to await another post. For now, we will describe it on the easier ring
For such a sequence
The Hankel matrix of
Let
sends the sequence
where
If
For the Fibonacci sequences
and therefore
There’s a lot of number theoretic and Galois-information encoded into the co-multiplication on
To see this we will describe the co-multiplication on
Here,
Finally, the co-algebra maps on the elements
That is, the co-multiplication on
Unlike the Fibonacci sequence, not every integral linear recursive sequence has an Hankel matrix with determinant
Reference: Richard G. Larson, Earl J. Taft, ‘The algebraic structure of linearly recursive sequences under Hadamard product’
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