Abstract

In this paper, we define the Gaussian Leonardo p numbers and we examine sum formula of the Gaussian Leonardo p numbers and then we define Q matrix of the Gaussian Leonardo p numbers. Also we give Binet formula of these numbers. We give the Gaussian Leonardo p numbers relations with the Leonardo p numbers and the Fibonacci p numbers.

1. Introduction

The Fibonacci sequence, a well-known example of an integer sequence, finds applications in mathematics computer science, and art. The Fibonacci and Lucas sequences are widely recognized as being among the most prominent integer sequences. In the realm complex numbers, Gaussian integers are characterized by having integer values for both their real and imaginary components. Gaussian integers, with their captivating properties, have drawn the attention of numerous researchers throughout history. Gauss’s groundbreaking contributions mathematics included the introduction of Gaussian integers in ¹. Complex Fibonacci numbers, introduced by Horadam, have found applications in various fields including number theory, algebra, and physics in ². The concepts of Gaussian Fibonacci and Gaussian Lucas numbers were brought forth by J.H. Jordan’s innovative research in ³ and he expanded the applicability of established relationships between Fibonacci sequences to Gaussian Fibonacci numbers. Pethe and Horadam introduced generalized form of Gaussian Fibonacci numbers, expanding the mathematical landscape in ⁴. Berzsenyi’s work marked a significant milestone in the extension of Fibonacci numbers to the complex plane ⁵. Furthermore, Stakhov and Rozin derived explicit Binet formulas for Fibonacci and Lucas p-numbers, extending the classical Binet formulas in ⁶. Tasci studied in ^{7, 8, 9, 10} complex Fibonacci p numbers, Gaussian Padovan and Gaussian Pell-Padovan sequences, Gaussian Mersenne numbers, Gauss balancing numbers, and Gauss Lucas-balancing numbers.

Fibonacci and Lucas numbers are defined recursively for n≥1:

, with the initial conditions and , with the initial conditions , respectively see ¹¹. Gaussian Fibonacci numbers are also ³ defined recursively: with the initial values

. It is clear that these numbers are closely related to Fibonacci numbers:, where . Similarly Gaussian Lucas numbers are defined ³ recursively for : with the initial conditions . Leonardo numbers are studied by Catarino and Burgers in ^{12, 13, 14}. They defined these numbers by the second order inhomogeneous recurrence relation :, with the initial conditions . Also, these numbers can be defined as : . Gaussian Leonardo numbers are studied by D.Tasci in ¹⁵. Gaussian Leonardo numbers are defined as , with the initial conditions . Some Gaussian Leonardo numbers: It is clear that Gaussian Leonardo numbers are closely related to Leonardo numbers: where denotes the nth Leonardo number. Also for it holds that .There are many important generalizations of Fibonacci numbers. The generalized Fibonacci p- numbers ¹⁶ examples of them and are defined by:

with the initial conditions The generalized Lucas p-numbers are defined by : with the initial conditions and where p=0,1,2 and . Let p be an integer the Gaussian Fibonacci p-numbers are defined ¹⁷ by the following recurrence relation

with the initial conditions and . It can be easily seen that where is the nth Fibonacci p-number. For any given integer p>0, the Leonardo p-sequence is defined ¹⁸ by the following non-homogenous relation :

with the initial values . The non-homogenous recurrence relation of Leonardo p numbers can be converted to the following homogenous recurrence relation :

Now we will define the Gaussian Leonardo p-numbers, we will give some results of these numbers.

2. Main Results

We start by giving the definition of the Gaussian Leonardo p-sequence.

Definition 2.1

For any given integer p >0, the Gaussian Leoanardo p –sequence is defined by the following non-homogenous relation :

with the initial values and .

It is clear that Gaussian Leonardo p-numbers are closely related to Leoanrdo p-numbers:

The non-homogenous recurrence relation of Gaussian Leonardo p-numbers can be converted to the following homogenous recurrence relation :

For p=1, we get Lemma 2.1 in ¹⁷

For p=1, p=2 and p=3 we give some of the numbers as follows

Theorem 2.1

For ,we have

where denotes the n-th Fibonacci p-numbers.

Proof 2.1

For , we have

and then we can get

Corollary 2.1

For p =1 , we get Theorem 2.1 in ¹⁷

Theorem 2.2

For , we have

where and denotes respectively the n-th Gaussian Leonardo p –numbers,Lucas p-numbers and Fibonacci p-numbers.

Proof 2.2

For ,we have

and then

we can get

Corollary 2.2

For p=1, we get theorem 2.2.b in ¹⁷

Theorem 2.3

The Binet formula of the Gaussian Leonardo p–sequence is

Corollary 2.3

For p=1, we get theorem 2.4 in ¹⁵

Theorem 2.4

For , we have

Proof 2.4

We have ,

and

Corallary 2.4

For p=1, we get theorem 2.6 in ¹⁵

Now we introduce the matrices that play the role of the Q-matrix.

Theorem 2.5

Let .Then

Proof 2.5

Theorem can be proved by mathematical induction on n.

Conclusion

This paper explores the properties and applications of Gaussian Leonardo p- numbers. Gaussian Leonardo p-numbers can be extended to Gaussian Lenardo p-polynomials.

Conflicts of Interest

The authors declare no conflict of interest.

ACKNOWLEDGEMENTS

The authors thank to the anonymous referees for his/her comments and valuable suggestions that improved the presentation of the manuscript.

References