On Gaussian Leonardo numbers

The Gaussian Leonardo sequence is a new sequence deﬁned in this study. Some identities for this new sequence are given. Some relations among the Gaussian Fibonacci numbers, Gaussian Lucas numbers, and Gaussian Leonardo numbers are also proven. Moreover, a matrix representation of the Gaussian Leonardo numbers is obtained.


Introduction
Sequences of integers have an important role in various fields, including computer science, physics, cryptology, and coding theory. The Fibonacci and Lucas sequences are among the most famous sequences of integers. A Gaussian integer is a complex number whose both imaginary and real parts are integers. The Gaussian integers have been investigated by many researchers. Gaussian integers were first considered by Gauss in [6]. Horadam introduced the complex Fibonacci numbers in [7]. Jordan introduced Gaussian Fibonacci and Gaussian Lucas numbers in [8] and he extended some well-known relations about Fibonacci sequences to Gaussian Fibonacci numbers. Pethe and Horadam investigated generalized Gaussian Fibonacci numbers in [9]. Berzsenyi extended the concept of Fibonacci numbers to the complex plane [2]. Furthermore, Taşcı studied in [10][11][12][13] 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 by recursively, for n ≥ 1: with the initial conditions F 0 = 0, F 1 = 1 and L n+1 = L n + L n−1 , with the initial conditions L 0 = 2, L 1 = 1, respectively (see [1]). Gaussian Fibonacci numbers GF n are also defined recursively: with the initial values GF 0 = i, GF 1 = 1. It is clear that these numbers are closely related to Fibonacci numbers: where i = √ −1. Similarly, Gaussian Lucas numbers GL n are defined recursively, for n ≥ 2: with the initial conditions GL 0 = 2 + i, GL 1 = 1 + 2i. Leonardo numbers are studied by Catarino and Burgers in [3][4][5]. They defined these numbers by the second order inhomogeneous recurrence relation: with the initial conditions Le 0 = Le 1 = 1. Also, these numbers can be defined as: The purpose of the present study is to introduce Gaussian Leonardo numbers and investigate their properties.

Gaussian Leonardo numbers
We begin with the definition of the Gaussian Leonardo numbers.
It is clear that Gaussian Leonardo numbers are closely related to Leonardo numbers: where Le n denotes the n-th Leonardo number.
Proof. Using Equations (5) and (7), we have The next result gives a relation between the Gaussian Leonardo and Gaussian Fibonacci numbers.
Theorem 2.1. For n ≥ 0, the following identity holds where GF n+1 denotes the (n + 1)-th Gaussian Fibonacci number and GLe n denotes the n-th Gaussian Leonardo number.
Proof. Consider Identities (2) and (7), together with where Le n denotes the n-th Leonardo number and F n+1 denotes the (n + 1)-th Fibonacci number. Then, we have Proof. In view of Theorem 2.1, we write Lemma 2.3. For n ≥ 1, the following identities hold: where GF n denotes n-th Gaussian Fibonacci number and GL n denotes n-th Gaussian Lucas number.
Proof. We prove Part (a) by the induction on n. Part (b) can be proved in a similar way. For n = 1, we have and the induction starts. Now, suppose that the desired identity holds for every k satisfying 1 < k ≤ n. Using GF n+1 = GF n + GF n−1 and the induction hypothesis, we have Thus, the required identity holds for n + 1, as desired.
The next result gives relationships among Gaussian Leonardo, Gaussian Lucas and Gaussian Fibonacci numbers. GLe Proof. Consider the Gaussian Leonardo sequence {GLe n } ∞ n=0 and let g(x) be its generating function, i.e., GLe n x n .

Theorem 2.4. Binet's formula for the Gaussian Leonardo numbers is
Proof. We know that the Binet formula for the Fibonacci numbers is It is easily seen that we have

Theorem 2.5. Cassini's Identity for the Gaussian Leonardo sequence is
Proof. Considering Theorem 2.1 and using and the result follows.