### Article

Kyungpook Mathematical Journal 2024; 64(3): 395-405

**Published online** September 30, 2024 https://doi.org/10.5666/KMJ.2024.64.3.395

Copyright © Kyungpook Mathematical Journal.

### The Exponential Representations of Pell and Its Generalized Matrix Sequences

Sukran Uygun

Department of Mathematics, Science and Art Faculty, Gaziantep University, Campus, 27310, Gaziantep, Turkey

e-mail : suygun@gantep.edu.tr

**Received**: June 10, 2023; **Revised**: March 13, 2024; **Accepted**: April 1, 2024

### Abstract

In this paper we define a matrix sequence called the Pell matrix sequence whose elements consist of Pell numbers. Using a positive parameter *k*, we generalize the Pell matrix sequence to a *k*-Pell matrix sequence and using two parameters *s*, *t* we generalize them to (*s*, *t*)-Pell matrix sequences. We give the basic properties of these matrix sequences. Then, using these properties we obtain exponential representations of the Pell matrix sequence and its generalizations in different ways.

**Keywords**: Pell numbers, matrix sequences, generalized sequences, exponential matrices

### 1. Introduction and Preliminaries

Sequences of positive integers have long been studied and many special integer sequences are known to have applications in different areas of science. Many researchers devote their attention to special sequences, such a Pell, Pell-Lucas, and Modified Pell sequences, which satisfy a second-order recurrence relation. Horadam studied various properties of Pell numbers and Pell polynomials. Ercolano found generating matrices for Pell sequences. Many mathematicians have looked at generalizations of Pell sequences one gets by adding one or two parameters to the recursion relation but not altering the initial conditions. Identities and *generalting* functions for the *k*-Pell numbers were established in [3]. The authors of [2] investigated (*s*,*t*)-Pell and (*s*,*t*)-Pell-Lucas sequences and their matrix representations. In [4], (*s*,*t*)-Pell and Pell-Lucas numbers are studied using matrix methods. In [5], the exponential representations of the Jacobsthal matrix sequences were found. In this paper we give the definitions of Pell sequence and its parametrized generalizations. Using the elements of the sequence, we establish matrix sequences for the integer sequences. We demonstrate the exponential matrices for the Pell matrix sequence and its generalizations by various methods.

As seen [1, 6], the recurrence relation with initial conditions for the Pell sequence is given as

The characteristic equation for the recurrence relation of the Pell sequence is

with roots

The sequence can be generalized using one parameter *k*, which is any positive integer. The *k*-Pell sequence

It has the characteristic equation

with roots

So, the following properties are established

The Binet formula for the *k*-Pell sequence with roots

As established in [4, 5], the two-parameter Pell sequence* *(*s*,*t*) *-*Pell sequence

where *s*,*t* are real numbers such that

with roots

The Binet formula for (*s*,*t*)-Pell numbers with the roots

### 2. Pell and Its Generalized Matrix Sequences

The Pell matrix sequence

The elements of Pell matrix sequence are the elements of Pell sequence such that

The *k*-Pell matrix sequence

The elements of *k*-Pell matrix sequence are the elements of *k*-Pell sequence such that

The

The elements of (*s*,*t*)-Pell matrix sequence are the elements of (*s*,*t*)-Pell sequence such that

**Lemma 2.1.** Assume *s*,*t* are real numbers such that *k* any positive integer, the following identities hold:

*Proof.* The proof is made by induction method. We want to prove the last equality that *k*=*n*+1:

If we choose *s*=*t*=1 in this equality, we get the first equality.

Similarly, If we choose *s*=1, *t*=*k*, we get the second equality.

**Lemma 2.2.** Assume *s*,*t* are real numbers such that *k* any positive integer, the following identities hold:

*Proof.* The proof is made by induction method. We want to prove the second equality that *i*=*n*+1:

The other proofs are made by using the same procedure.

### 3. The Exponential Representations of Pell Matrix Sequences

In this section, we want to present the exponential representations of the *nth* element of Pell matrix sequence and the *nth* element of generalized Pell matrix sequences. If a function *f*(*z*) of a complex variable *z* has a Maclaurin series expansion *A* is square and each of its eigenvalues has absolute value less than *R*. In such a case, *f*(*A*) is defined as

**Theorem 3.1.** *For any integer *

where *nth* element of (*s*,*t*)-Pell matrix sequence using the *nth* power of first element of (*s*,*t*)-Pell matrix sequence. Similarly, the exponential representation of the *nth* element of *k*-Pell matrix sequence is

and the exponential representation of the *nth* element of the Pell matrix sequence is

*Proof.* The eigenvalues of

By these equations, the values of

Applying the Maclaurin series expansion of

and

If the results are combined

If we choose *s*=*t*=1, we can apply this result for classic Pell matrix sequence defined in (2.1). The eigenvalues of

where *nth* element of the classic Pell matrix sequence.

If we choose *s*=1, *k*-Pell matrix sequence defined in (2.2). The eigenvalues of

where *nth* element of the *k*-Pell matrix sequence.

The following theorem shows us a second way for expressing the exponential representation of the *nth* element of (*s*,*t*)-Pell matrix sequence.

**Theorem 3.2.** *For any integer *

*where U is an invertible matrix and*

*This result has more advantage for finding the exponential representations of the nth element of (s,t)-Pell matrix sequence. Because we only need the elements of the sequence *

*Proof.* Because of the eigenvalues of *U* matrix such that

Therefore, the exponential form is

Then, we obtain

If we choose *s*=*t*=1, we can apply this result for classic Pell matrix sequence as

where

If we choose *s*=1, *k*-Pell matrix sequence. The eigenvalues of matrix are

where

**Theorem 3.3.** *For *

*Proof.* By using Lemma 2.2, there is an invertible *U* matrix such that

By using the properties of the exponential matrix, we have

We also obtain

and

By combining the results, the proof is completed.

Exponential representation of (2*n*)*th* *k*-Pell and (*s*,*t*)-Pell matrix sequences can be obtained by using the same procedure. We give the results as

**Theorem 3.4.** *For *

*Proof.* The eigenvalues of

By these equations, the values of

If we choose *s*=*t*=1, we can apply this result for the classic Pell matrix sequence

If we choose *s*=1, *k*-Pell matrix sequence

### 4. Conclusion

The exponential representations of the Pell matrix sequence and its generalized matrix sequences are investigated in this study. The elements of the sequences are

### References

- A. F. Horadam.
*Pell Identities*, Fibonacci Quart.,**9(3)**(1971), 245-252. - H. H. Gulec and N. Taskara.
*On the (s,t)-Pell and (s,t)-Pell-Lucas sequences and their matrix representations*, Appl. Math. Lett.,**25(10)**(2012), 1554-1559. - P. Catarino and P. Vasco.
*On Some Identities and Generating Functions for k-Pell Numbers*, Int. J. Math. Anal.,**7(38)**(2013), 1877-1884. - S. Srisawat and W. Sriprad.
*On the (s,t)-Pell and Pell-Lucas Numbers by Matrix Methods*, Ann. Math. Inform.,**46**(2016), 195-204. - S. Uygun and E. Owusu.
*The Exponential Representations of Jacobsthal Matrix Sequences*, J. Math. Anal.,**7(5)**(2016), 140-146. - T. Koshy,
*Pell and Pell-Lucas Numbers with Applications*, Springer, Berlin, 2014.