We apply the binomial transforms to Padovan and Perrin matrix sequences. Also, the Binet formulas, summations, and generating functions of these transforms are found by recurrence relations. Finally, we illustrate the relations between these transforms by deriving new formulas.

There are so many studies in the literature that are concernes about the special number sequences such as Fibonacci, Lucas, Pell, Jacobsthal, Padovan, and Perrin (see, e.g., [

Although the study of Perrin numbers started in the beginning of the 19th. century under different names, the master study was published in 2006 by Shannon et al. in [

On the other hand, the matrix sequences have taken so much interest for different types of numbers (cf. [

In addition, some matrix based transforms can be introduced for a given sequence. Binomial transform is one of these transforms, and there are also other ones such as rising and falling binomial transforms (see [

Motivated by [

Now, we give some preliminaries related to our study. Given an integer sequence

In [

Let one considers

In this section, we will mainly focus on binomial transforms of Padovan and Perrin matrix sequences to get some important results. In fact, as a middle step, we will also present the recurrence relations, Binet formulas, summations, and generating functions.

Let

the binomial transform of the Padovan matrix sequence is

the binomial transform of the Perrin matrix sequence is

We note that, from Definition

The following lemma will be the key of the proof of the next theorems.

For

Firstly, in here we will just prove (i), since (ii) can be thought in the same manner with (i).

(i) By using Definition

From the previous lemma, note that

For

recurrence relation of sequences

recurrence relation of sequences

Similarly for the proof of the previous theorem, only the first case (i) will be proved. We will omit the other cases since the proofs will not be different.

(i) By considering the right-hand side of equality in (i) and Definition

By taking, account equality

From Lemma

On the other hand, by using (

The characteristic equation of sequences

Sums of sequences

(i) By considering (

Now, if we take

Afterwards, by taking into account (

(ii) The proof of the binomial transform of Perrin matrix sequences can be seen by taking into account (

The generating functions of the binomial transforms for

We omit Padovan case since the proof will be quite similar.

Assume that

Hence, the result is obtained.

In this section, we present the relationship between these binomial transforms.

For

(i) From Definition

By considering Proposition

By taking into account Vandermonde’s identity

(ii) Here, we will just show that the truthness of the equality

(iii) By considering (

From (

Now, by taking into account again (

The final part of the proof can be seen similarly as in the proof of (iii).

The properties of the transforms

We will omit the proof of (ii) and (iii), since it is quite similar to (i). Therefore, by considering Definition

For

By considering Definition

By choosing

The following equalities are held:

The authors declare that there is no conflict of interests regarding the publication of this paper.