Hybrid Quaternions of Leonardo

In this article, we intend to investigate the Leonardo sequence presenting the hybrid Leonardo quaternions. To explore Hybrid Quaternions of Leonardo, the priori, sequence of Leonardo, quaternions and hybrid numbers were presented. Soon after, its recurrence, characteristic equation, its relation with the Fibonacci quaternions, generating function, Binet’s formula, as well as its extension to non-positive integer indices were developed. Finally, identities involving Leonardo’s hybrid quaternions are presented.

Keywords:
Leonardo sequence; hybrid Leonardo quaternions; hybrid numbers

Authorship

M. C. dos S. MANGUEIRA ^**Corresponding author: Milena Carolina Santos Mangueira - E-mail: milenacarolina24@gmail.com

Federal Institute of Education, Science, Technology of Ceará - IFCE, Department of Mathematics, Treze of Maio, 2081, 60040-531, Fortaleza, CE, Brazil - E-mail: milenacarolina24@gmail.comFederal Institute of Education, Science, Technology of CearáBrazilFortaleza, CE, BrazilFederal Institute of Education, Science, Technology of Ceará - IFCE, Department of Mathematics, Treze of Maio, 2081, 60040-531, Fortaleza, CE, Brazil - E-mail: milenacarolina24@gmail.com

http://orcid.org/0000-0003-2690-7033

F. R. V. ALVES

Federal Institute of Education, Science, Technology of Ceará - IFCE, Department of Mathematics, Treze of Maio, 2081, 60040-531, Fortaleza, CE, Brazil - E-mail: fregis@ifce.edu.brFederal Institute of Education, Science, Technology of CearáBrazilFortaleza, CE, BrazilFederal Institute of Education, Science, Technology of Ceará - IFCE, Department of Mathematics, Treze of Maio, 2081, 60040-531, Fortaleza, CE, Brazil - E-mail: fregis@ifce.edu.br

http://orcid.org/0000-0003-3710-1561

P. M. M. C. CATARINO

University of Trás-os-Montes, Department of Mathematics, Portugal - E-mail: pcatarino23@gmail.comUniversity of Trás-os-MontesPortugalPortugalUniversity of Trás-os-Montes, Department of Mathematics, Portugal - E-mail: pcatarino23@gmail.com

http://orcid.org/0000-0001-6917-5093

*Corresponding author: Milena Carolina Santos Mangueira - E-mail: milenacarolina24@gmail.com

SCIMAGO INSTITUTIONS RANKINGS

Tables | Formulas

Tables (1)
Formulas (38)

Thumbnail

Table 1:
First terms of Leonardo’s hybrids and quaternions

(1.1)

(1.2)

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(3.1)

Table 1: First terms of Leonardo’s hybrids and quaternions

n	HLe _n	QLe _n
0	1 + i + 3ε + 5h	1 + i + 3 j + 5k
1	1 + 3i + 5ε + 9h	1 + 3i + 5 j + 9k
2	3 + 5i + 9ε + 15h	3 + 5i + 9 j + 15k
3	5 + 9i + 15ε + 25h	5 + 9i + 15 j + 25k
4	9 + 15i + 25ε + 41h	9 + 15i + 25 j + 41k
5	15 + 25i + 41ε + 67h	15 + 25i + 41 j + 67k
6	25 + 41i + 67ε + 109h	25 + 41i + 67 j + 109k
⁝	⁝	⁝

(1.1)

L e_{n} = L e_{n - 1} + L e_{n - 2} + 1, n \geq 2,

(1.2)

L e_{n} - L e_{n + 1} = L e_{n - 1} + L e_{n - 2} + 1 - L e_{n} - L e_{n - 1} - 1 L e_{n + 1} = 2 L e_{n} - L e_{n - 2}

L e_{n} = 2 F_{n + 1} - 1 .

q = a + b i + c j + d k

K = \{z = a + b i + c ε + d h : a, b, c, d \in R, i^{2} = - 1, ε^{2} = 0, h^{2} = 1, i h = - h i = ε + i\}

z = a - b i - c ε - d h

C (z) = z z = z z = a^{2} + {(b - c)}^{2} - c^{2} - d^{2} = a^{2} + b^{2} - 2 b c - d^{2}

H L e_{n} = L e_{n} + L e_{n + 1} i + L e_{n + 2} ε + L e_{n + 3} h .

(2.1)

H L e_{n} = H L e_{n - 1} + H L e_{n - 2} + (1 + i + ε + h),

Q L e_{n} = L e_{n} + L e_{n + 1} i + L e_{n + 2} j + L e_{n + 3} k .

(2.2)

Q L e_{n} = Q L e_{n - 1} + Q L e_{n - 2} + (1 + i + j + k),

{\tilde{L e}}_{n} = H L e_{n} + H L e_{n + 1} i + H L e_{n + 2} j + H L e_{n + 3} k .

{\tilde{L e}}_{n} = (L e_{n} + L e_{n + 1} i + L e_{n + 2} ε + L e_{n + 3} h) + (L e_{n + 1} + L e_{n + 2} i + L e_{n + 3} ε + L e_{n + 4} h) i + (L e_{n + 3} + L e_{n + 4} i + L e_{n + 5} ε + L e_{n + 6} h) k = {\hat{H L e}}_{n} + {\hat{H L e}}_{n + 1} i + {\hat{H L e}}_{n + 2} ε + {\hat{H L e}}_{n + 3} h

(2.3)

{\tilde{L e}}_{n + 1} = 2 {\tilde{L e}}_{n} - {\tilde{L e}}_{n - 2},

{\tilde{L e}}_{- n} = 2 {\tilde{L e}}_{- n + 2} - {\tilde{L e}}_{- n + 3} .

(2.4)

{\tilde{L e}}_{n + 1} = {\tilde{L e}}_{n} + {\tilde{L e}}_{n - 1} + [Θ + Θ i + Θ j + Θ k],

{\tilde{L e}}_{n} + {\tilde{L e}}_{n - 1} + [Θ + Θ i + Θ j + Θ k] = H L e_{n} + H L e_{n + 1} i + H L e_{n + 2} j + H L e_{n + 3} k + H L e_{n - 1} + H L e_{n} i + H L e_{n + 1} j + H L e_{n + 2} k + [Θ + Θ i + Θ j + Θ k] = (H L e_{n} + H L e_{n - 1} + Θ) + (H L e_{n + 1} + H L e_{n} + Θ) i + (H L e_{n + 2} + H L e_{n + 1} + Θ) j + (H L e_{n + 3} + H L e_{n + 2} + Θ) k + H L e_{n + 1} + H L e_{n + 2} i + H L e_{n + 3} j + H L e_{n + 4} k = {\tilde{L e}}_{n + 1}

(2.5)

{\tilde{L e}}_{n} = 2 {\tilde{F}}_{n + 1} - [Θ + Θ i + Θ j + Θ k],

{\tilde{L e}}_{0} = 2 {\tilde{F}}_{1} - [Θ + Θ i + Θ j + Θ k] = 2 (H F_{1} + H F_{2} i + H F_{3} j + H F_{4} k) - [Θ + Θ i + Θ j + Θ k] = (2 H F_{1} - Θ) + (2 H F_{2} - Θ) i + (2 H F_{3} - Θ) j + (2 H F_{4} - Θ) k = (1 + i + 3 ε + 5 h) + (1 + 3 i + 5 ε + 9 h) i + (2 + 5 i + 9 ε + 15 h) j + (5 + 9 i + 15 ε + 25 h) k = H L e_{0} + H L e_{1} i + H L e_{2} j + H L e_{3} k = {\tilde{L e}}_{0}

{\tilde{L e}}_{t + 1} = {\tilde{L e}}_{t} + {\tilde{L e}}_{t - 1} + [Θ + Θ i + Θ j + Θ k] = 2 {\tilde{F}}_{t + 1} - [Θ + Θ i + Θ j + Θ k] + 2 {\tilde{F}}_{t} - [Θ + Θ i + Θ j + Θ k] + [Θ + Θ i + Θ j + Θ k] = 2 ({\tilde{F}}_{t + 1} + {\tilde{F}}_{t}) - [Θ + Θ i + Θ j + Θ k] = 2 {\tilde{F}}_{t + 2} - [Θ + Θ i + Θ j + Θ k]

\frac{{\tilde{L e}}_{n + 1}}{{\tilde{L e}}_{n}} = 2 - \frac{{\tilde{L e}}_{n - 2}}{{\tilde{L e}}_{n}} .

\lim_{n \to \infty} x_{n} = 2 - \frac{1}{l i m_{n \to \infty} x_{n - 1} \cdot l i m_{n \to \infty} x_{n - 2}} x = 2 - \frac{1}{x^{2}} x^{3} - 2 x^{2} + 1 = 0,

G_{{\tilde{L e}}_{n}} (t) = \frac{({\tilde{L e}}_{0} + {\tilde{L e}}_{1} t) (1 - 2 t) + {\tilde{L e}}_{2} t^{2}}{1 - 2 t + t^{3}}

G_{{\tilde{L e}}_{n}} (t) = \sum_{n = 0}^{\infty} {\tilde{L e}}_{n} t^{n}

G_{{\tilde{L e}}_{n}} (t) = {\tilde{L e}}_{0} + {\tilde{L e}}_{1} t + {\tilde{L e}}_{2} t^{2} + \sum_{n = 3}^{\infty} {\tilde{L e}}_{n} t^{n} = {\tilde{L e}}_{0} + {\tilde{L e}}_{1} t + {\tilde{L e}}_{2} t^{2} + \sum_{n = 3}^{\infty} (2 {\tilde{L e}}_{n - 1} - {\tilde{L e}}_{n - 3}) t^{n} = {\tilde{L e}}_{0} + {\tilde{L e}}_{1} t + {\tilde{L e}}_{2} t^{2} + 2 t \sum_{n = 3}^{\infty} {\tilde{L e}}_{n - 1} t^{n - 1} - t^{3} \sum_{n = 3}^{\infty} {\tilde{L e}}_{n - 3} t^{n - 3} = {\tilde{L e}}_{0} + {\tilde{L e}}_{1} t + {\tilde{L e}}_{2} t^{2} + 2 t \sum_{n = 0}^{\infty} ({\tilde{L e}}_{n} t^{n} - {\tilde{L e}}_{0} + {\tilde{L e}}_{1} t) - t^{3} \sum_{n = 0}^{\infty} {\tilde{L e}}_{n} t^{n} = {\tilde{L e}}_{0} + {\tilde{L e}}_{1} t + {\tilde{L e}}_{2} t^{2} - 2 {\tilde{L e}}_{0} t - 2 {\tilde{L e}}_{1} t^{2} + 2 t \sum_{n = 0}^{\infty} {\tilde{L e}}_{n} t^{n} - t^{3} \sum_{n = 0}^{\infty} {\tilde{L e}}_{n} t^{n} = {\tilde{L e}}_{0} + {\tilde{L e}}_{1} t + {\tilde{L e}}_{2} t^{2} - 2 {\tilde{L e}}_{0} t - 2 {\tilde{L e}}_{1} t^{2} + 2 t G_{{\tilde{L e}}_{n}} - t^{3} G_{{\tilde{L e}}_{n}}

G_{{\tilde{L e}}_{n}} (t) - 2 t G_{{\tilde{L e}}_{n}} + t^{3} G_{{\tilde{L e}}_{n}} = {\tilde{L e}}_{0} + {\tilde{L e}}_{1} t + {\tilde{L e}}_{2} t - 2 {\tilde{L e}}_{0} t - 2 {\tilde{L e}}_{1} t^{2} G_{{\tilde{L e}}_{n}} (t) (1 - 2 t + t^{3}) = {\tilde{L e}}_{0} (1 - 2 t) + {\tilde{L e}}_{1} t (1 - 2 t) + {\tilde{L e}}_{2} t^{2} G_{{\tilde{L e}}_{n}} (t) = \frac{({\tilde{L e}}_{0} + {\tilde{L e}}_{1} t) (1 - 2 t) + {\tilde{L e}}_{2} t^{2}}{1 - 2 t + t^{3}}

{\tilde{L e}}_{n} = A x_{1}^{n} + B x_{2}^{n} + C_{3}^{n},

A = \frac{({\tilde{L e}}_{2}) + (- x_{2} - x_{3}) ({\tilde{L e}}_{1}) + x_{2} x_{3} ({\tilde{L e}}_{0})}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}}, B = \frac{({\tilde{L e}}_{2}) + (- x_{1} - x_{3}) ({\tilde{L e}}_{1}) + x_{1} x_{3} ({\tilde{L e}}_{0})}{x_{2}^{2} - x_{2} x_{3} - x_{1} x_{2} + x_{1} x_{3}}, C = \frac{({\tilde{L e}}_{2}) + (- x_{1} - x_{2}) ({\tilde{L e}}_{1}) + x_{1} x_{2} ({\tilde{L e}}_{0})}{x_{3}^{2} - x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3}} .

\{\begin{cases} A + B + C = {\tilde{L e}}_{0} \\ A x_{1} + B x_{2} + C x_{3} = {\tilde{L e}}_{1} \\ A x_{1}^{2} + B x_{2}^{2} + C x_{3}^{2} = {\tilde{L e}}_{2} \end{cases}

A = \frac{({\tilde{L e}}_{2}) + (- x_{2} - x_{3}) ({\tilde{L e}}_{1}) + x_{2} x_{3} ({\tilde{L e}}_{0})}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}}, B = \frac{({\tilde{L e}}_{2}) + (- x_{1} - x_{3}) ({\tilde{L e}}_{1}) + x_{1} x_{3} ({\tilde{L e}}_{0})}{x_{2}^{2} - x_{2} x_{3} - x_{1} x_{2} + x_{1} x_{3}}, C = \frac{({\tilde{L e}}_{2}) + (- x_{1} - x_{2}) ({\tilde{L e}}_{1}) + x_{1} x_{2} ({\tilde{L e}}_{0})}{x_{3}^{2} - x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3}} .

(3.1)

\sum_{j = 0}^{n} {\tilde{L e}}_{j} = {\tilde{L e}}_{n + 2} - [n + 1 + (Θ + Θ i + Θ j + Θ k)],

\sum_{j = 0}^{n} {\tilde{L e}}_{j} = \sum_{j = 0}^{n} [2 {\tilde{F}}_{j + 1} - (Θ + Θ i + Θ j + Θ k)] = 2 \sum_{j = 0}^{n} {\tilde{F}}_{j + 1} - [n + (Θ + Θ i + Θ j + Θ k)] = 2 \sum_{j = 0}^{n + 1} {\tilde{F}}_{j} - [n + (Θ + Θ i + Θ j + Θ k)] = 2 ({\tilde{F}}_{n + 2} + {\tilde{F}}_{n + 1} - 1) - [n + (Θ + Θ i + Θ j + Θ k)] = 2 ({\tilde{F}}_{n + 3} - 1) - [n + (Θ + Θ i + Θ j + Θ k)] = {\tilde{L e}}_{n + 2} - [n + 1 + (Θ + Θ i + Θ j + Θ k)]

\sum_{j = 0}^{n} {\tilde{L e}}_{2 j} = {\tilde{L e}}_{2 n + 1} - n

\sum_{j = 0}^{n} {\tilde{L e}}_{2 j} = \sum_{j = 0}^{n} [2 {\tilde{F}}_{2 j + 1} - (Θ + Θ i + Θ j + Θ k)] = 2 \sum_{j = 0}^{n} {\tilde{F}}_{2 j + 1} - [n + (Θ + Θ i + Θ j + Θ k)] = 2 {\tilde{F}}_{2 n + 2} - [n + (Θ + Θ i + Θ j + Θ k)] = [2 {\tilde{F}}_{(2 n + 1) + 1} - (Θ + Θ i + Θ j + Θ k)] - n = {\tilde{L e}}_{2 n + 1} - n

\sum_{j = 0}^{n} {\tilde{L e}}_{2 j + 1} = Q L e_{2 n + 2} - (n + 2)

\sum_{j = 0}^{n} {\tilde{L e}}_{2 j + 1} = \sum_{j = 0}^{n} [2 {\tilde{F}}_{2 j + 2} - (Θ + Θ i + Θ j + Θ k)] = 2 \sum_{j = 0}^{n} {\tilde{F}}_{2 j} - [n + (Θ + Θ i + Θ j + Θ k)] = 2 {\tilde{F}}_{2 n + 3} - [n + (Θ + Θ i + Θ j + Θ k)] = [2 {\tilde{F}}_{(2 n + 2) + 1} - (Θ + Θ i + Θ j + Θ k)] - (n + 2) = {\tilde{L e}}_{2 n + 1} - (n + 2)

\sum_{j = 0}^{n} ({\tilde{F}}_{j} + {\tilde{L e}}_{j}) = {\tilde{F}}_{n + 2} + {\tilde{L e}}_{n + 2} - [(n + 1) + 2 (Θ + Θ i + Θ j + Θ k)],

\sum_{j = 0}^{n} ({\tilde{F}}_{j} + {\tilde{L e}}_{j}) = \sum_{j = 0}^{n} {\tilde{F}}_{j} + \sum_{j = 0}^{n} {\tilde{L e}}_{j} = {\tilde{F}}_{n + 1} + {\tilde{F}}_{n} - (Θ + Θ i + Θ j + Θ k) + {\tilde{L e}}_{n + 2} - [n + 1 + (Θ + Θ i + Θ j + Θ k)] = {\tilde{F}}_{n + 2} + {\tilde{L e}}_{n + 2} - [(n + 1) + 2 (Θ + Θ i + Θ j + Θ k)]

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[1] *Corresponding author: Milena Carolina Santos Mangueira - E-mail: milenacarolina24@gmail.com