Hadamard product and binomial convolution of linear recurrences

Hi everyone!

Let be a ring, and be linear recurrence sequences, such that

In some applications, the following two sequences arise:

Today I'd like to write about the framework that allows to prove that both the sequences defined above are also linear recurrences. It would also allow to compute their characteristic polynomials in , which is optimal as their degrees are in both cases.

Umbral calculus

Generally, a linear recurrence can be described and analyzed with the help of the linear functional such that

For such functional, when is a multiple of the characteristic polynomial of . The existence of the characteristic polynomial is the criterion of being a linear recurrence. So, we need to prove that there is such a polynomial for defined above.

Joint umbral calculus

To analyze joint properties of and , we define a linear functional such that

Similarly to the case of a single recurrence, whenever is a linear combination of and , where

are the characteristic polynomials of and . In other words, whenever lies in the ideal .

Composed sum

For the binomial convolution let , then

To show that is a linear recurrence obeying to the rule

it is sufficient to show that there is a characteristics polynomial such that .

Assume that is an integral domain. Then the polynomial exists and can be defined explicitly as

where

The fact that is proven as follows:

In the sum above, there are summands, each of them is divisible by either or , so .

The polynomial defined above is called the composed sum of and .

Composed product

Now the question is, how to prove that the Hadamard product is a linear recurrence?

Using similar logic as above, one would define and then look for . Let

This one is a bit trickier to prove. Let's start with :

Rewriting it in the same way for arbitrary and , we get

Then the same logic applies as to in the binomial convolution case.

The polynomial defined above is called the composed product of and .

Computing composed products and sums

Let be the sum of -th powers of all and be the sum of -th powers of all , that is

The roots of the composed sum are for all and and of the composed product are , from which we can see that

So, if we're able to transform from to , then from to , compute the transforms above on them and then recover the characteristics polynomials from the result, it would solve the problem.

Next thing we should note is that the generating function of is

It can be further expanded as

where

is the reversed characteristic polynomial of . Its log-derivative is indeed

Finally, to inverse this transform, we could make use of the fact that , hence for

it holds that

The resulting has degree , so only terms of and are needed and they may be computed in .