Some properties and Vajda theorems of split dual Fibonacci and split dual Lucas octonions

1.   Introduction

Quaternions and octonions were described by Sir William R. Hamilton in 1843 and Cayley-Dickson in 1845, respectively. In 1873, Clifford extended real numbers to dual numbers ^[16]. For the real numbers $a$ and $a^{\ast}$, the form of the dual number $d$ is

where $\varepsilon$ is the dual unit and

The Binet formulas for the Fibonacci and Lucas numbers are

where $\alpha = \frac{1+\sqrt{5}}{2}$ and $\beta = \frac{1-\sqrt{5}}{2}$ are the roots of the characteristics equation

The positive root $\alpha$ is known as golden ratio. The quaternions were introduced by Horadam ^[3]. Some authors studied Fibonacci or split Fibonacci quaternions ^{[1,2,4,5,6,8,10,11,12,17,18]}.

Keçilioğlu and Akkuş gave the Fibonacci and Lucas octonions and their Binet formulas ^[7]. Halici also worked on dual Fibonacci octonions in ^[9] and obtained the Binet formula with its generator function. Ünal et al. investigated properties of dual Fibonacci and dual Lucas octonions ^[14]. In addition, Bilgici et al. obtained generalized Fibonacci and Lucas octonions, and gave a lot of identities to them ^[15].

In ^[13], Akkuş and Keçilioğlu took the form

of a pair of quaternions, shown as $(a, b)$, with a new imaginary unit $l$ and defined the product rule as

where $\lambda^{2} = 1.$ When $\lambda = -1$ and $\lambda = 1$ above, octonions and split octonions are obtained, respectively. Split octonions form an eight-dimensional algebra on real numbers. The difference from standard octonions is that it contains elements that are different from zero. Accordingly, let us show the split dual octonion algebra on real numbers with $\widetilde{\widetilde{O}}.$ In addition, Akkuş and Keçilioğlu have given the $n$th split Fibonacci and $n$th split Lucas octonions in the form of

and

where $F_{n}$ is $n$th Fibonacci number, $L_{n}$ is $n$th Lucas number ^[13]. If we take $a = 1, b = 1$ and $c = -1$ in the generalized octonion multiplication table in ^[15], we get the split octonion table (see Table 1).

We define split dual Fibonacci and split dual Lucas numbers by

and

respectively. In this paper, following Keçilioğlu and Akkuş, and Ünal et al., we define the split dual Fibonacci and split dual Lucas octonions over the split dual octonion algebra $\widetilde{\widetilde{O}}.$ The $n$th split dual Fibonacci octonion $SDFO_{n}$ is

and the $n$th split dual Lucas octonion $SDLO_{n}$ is

where $\tilde{F}_{n}$ is $n$th split dual Fibonacci number and $\tilde{L}_{n}$ is $n$th split dual Lucas number. By using Eq $\left(1.1\right)$, we obtain

where $SFO_{n} = { \sum\limits_{s = 0}^{7}} F_{n+s}e_{s}$ is split Fibonacci octonion. Similarly, by using Eq $\left(1.2\right)$, we have

where $SLO_{n} = { \sum\limits_{s = 0}^{7}} L_{n+s}e_{s}$ is split Lucas octonion.

2.   Binet formulas for split dual Fibonacci and Lucas octonions

In this section, we obtain Binet formulas and Vajda theorems for $SDFO_{n}$ and $SDLO_{n}$. There are three well-known identities for split dual Fibonacci and Lucas numbers, namely, Catalan's, Cassini's and d'Ocagne's identities. These types of identities for split dual Fibonacci and Lucas octonions can be obtained by using the Vajda theorems. The following theorem gives the Binet formulas for these octonions.

Theorem 1. For $n\geq0,$ the $n$th split dual Fibonacci octonion is

and $n$th split dual Lucas octonion is

where $\alpha^{\prime} = \left(1+\varepsilon\alpha\right) { \sum\limits_{s = 0}^{7}} \alpha^{s}e_{s}$ and $\beta^{\prime} = \left(1+\varepsilon\beta\right) { \sum\limits_{s = 0}^{7}} \beta^{s}e_{s}.$

Proof. In ^[13], the Binet formulas for the split Fibonacci and split Lucas octonions are as follows:

and

where $\alpha^{\ast} = { \sum\limits_{s = 0}^{7}} \alpha^{s}e_{s}$ and $\beta^{\ast} = { \sum\limits_{s = 0}^{7}} \beta^{s}e_{s}.$

By using from the Eq $(2.1)$, we get the Eq $(1.3)$. We know that

When the Binet formula of split Fibonacci octonion is used in this last equation, we get

If the expressions $\alpha^{\ast}$ and $\beta^{\ast}$ given in ^[13] are used,

is obtained. By using similar method, we get Binet formula of split dual Lucas octonion.

Using Binet's formulas, we can easily derive the identities between split dual Fibonacci and split dual Lucas octonions.

Now, we give some useful identities that play very important roles throughout the paper for calculations.

Lemma 1. We have obtained followings

where

Proof. The proof is completed when $a = b = 1$ and $c = -1$ in Lemma 2 in ^[15].

Now we give Vajda theorems for split dual Fibonacci and split dual Lucas octonions that could be reduced to some special cases.

Theorem 2. For any integers $n, r$ and $s$, we get

and

Proof. We make proof for split dual Fibonacci octnions. We can write

We calculate the following expression

and we get

If $s+1$ is written instead of $s$ in Eq $\left(2.3\right)$, then

is obtained. If $n+1$ is written instead of $n$ and $s-1$ is written instead of $s$ in equation given above, then we get

In this case,

and so,

Similarly, we can prove the Vajda theorem for split dual Lucas octonions.

From Vajda theorems, we also have the following special identities for these octonions:

Corollary 1. If $-r$ is taken instead of $s$ in the Vajda theorem, Catalan's identities are obtained for split dual Fibonacci and split dual Lucas octonions, respectively:

and

Corollary 2. If $-r$ is taken instead of $s$ and then $1$ is taken instead of $r$ in the Vajda theorem, Cassini's identities are obtained for split dual Fibonacci and split dual Lucas octonions, respectively :

and

where $F_{0} = 0, F_{1} = 1$ and $F_{2} = 1.$

Corollary 3. If $m$ is taken instead of $n$, $n-m$ is taken instead of $s$ and then $1$ is taken instead of $r$ in the Vajda theorem, d'Ocagne's identities are obtained for split dual Fibonacci and split dual Lucas octonions, respectively:

and

3.   Some results for split dual Fibonacci and split dual Lucas octonions

In this section, after deriving famous three identities Catalan's, Cassini's and d'Ocagne's by using the Vajda theorems, we present some other identities for the split dual Fibonacci and split dual Lucas octonions.

Theorem 3. Split dual Fibonacci octonions satisfy the following identities;

i) $SDFO_{m+n}+\left(-1\right) ^{n}SDFO_{m-n} = SDFO_{m}L_{n},$

ii) $SDFO_{n+r}F_{n+r}-SDFO_{n-r}F_{n-r} = F_{2r}SDFO_{2n},$

iii) $SDFO_{n}SDFO_{m}-SDFO_{m}SDFO_{n} = 2\left(-1\right) ^{m+1}\lambda F_{n-m}\left(1+\varepsilon\right).$

Proof. We prove only the first identity of this theorem. We need the Binet formulas for the split dual Fibonacci octonions.

i) $SDFO_{m+n}+\left(-1\right) ^{n}SDFO_{m-n}$

$= \left(\tilde{Q}_{m+n}+\varepsilon\tilde{Q}_{m+n+1}\right) +\left(\alpha\beta\right) ^{n}\left(\tilde{Q}_{m-n}+\varepsilon\tilde{Q} _{m-n+1}\right)$

$= \left(\alpha^{n}+\beta^{n}\right) \left(\tilde{Q}_{m}+\varepsilon \tilde{Q}_{m+1}\right)$ $= \left(\alpha^{n}+\beta^{n}\right) SDFO_{m} = SDFO_{m}L_{n}.$

The second and third identities can be proved similarly.

Theorem 4. Split dual Lucas octonions satisfy the following identities;

i) $SDLO_{n+r}L_{n+r}-SDLO_{n-r}L_{n-r} = 5F_{2r}SDFO_{2n},$

ii) $SDLO_{n+r}L_{n+r}+SDLO_{n-r}L_{n-r} = L_{2r}SDLO_{2n}+2\left(-1\right) ^{n+r}SDLO_{0},$

iii) $SDLO_{n}SDLO_{m}-SDLO_{m}SDLO_{n} = 10\left(-1\right) ^{m}\lambda F_{n-m}\left(1+\varepsilon\right)$.

Proof. We prove only the first identity of this theorem. We need the Binet formulas for the split dual Lucas octonions

i) $SDLO_{n+r}L_{n+r}-SDLO_{n-r}L_{n-r}$ $= \left(\tilde{P}_{n+r}+\varepsilon\tilde{P}_{n+r+1}\right) L_{n+r}-\left(\tilde{P}_{n-r}+\varepsilon\tilde{P}_{n-r+1}\right) L_{n-r}$

$= \left(\alpha^{2r}-\beta^{2r}\right) \left(\alpha^{\ast}\alpha ^{2n}-\beta^{\ast}\beta^{2n}\right) +\varepsilon\left(\alpha^{2r}-\beta^{2r}\right) \left(\alpha^{\ast }\alpha^{2n+1}-\beta^{\ast}\beta^{2n+1}\right)$

$= \left(\alpha-\beta\right) ^{2}\left(\dfrac{\alpha^{2r}-\beta^{2r} }{\alpha-\beta}\right) \left[ \begin{array} [c]{c} \left(\dfrac{\alpha^{\ast}\alpha^{2n}-\beta^{\ast}\beta^{2n}}{\alpha-\beta }\right) +\varepsilon\left(\dfrac{\alpha^{\ast}\alpha^{2n+1}-\beta^{\ast}\beta ^{2n+1}}{\alpha-\beta}\right) \end{array} \right]$

$= 5F_{2r}\left(\tilde{Q}_{2n}+\varepsilon\tilde{Q}_{2n+1}\right) = 5F_{2r}SDFO_{2n}.$

The other two identities can be proved similar to the theorem proved by using split dual Fibonacci octonion.

Theorem 5. Split dual Fibonacci and split dual Lucas octonions satisfy the following identities;

i) $SDFO_{m}SDLO_{n}-SDFO_{n}SDLO_{m} = 2\left(-1\right) ^{m+1}SPO_{0}F_{n-m}\left(1+\varepsilon\right),$

ii) $SDFO_{m}SDLO_{n}-SDLO_{m}SDFO_{n} = 2\left(-1\right) ^{m+1}\left(SPO_{0}F_{n-m}-\lambda L_{n-m}\right) \left(1+\varepsilon\right),$

iii) $SDLO_{n+r}SDFO_{n+s}-SDLO_{n+s}SDFO_{n+r} = 2\left(-1\right) ^{n+r}SPO_{0}F_{s-r}\left(1+\varepsilon\right),$\newline\newline iv) $SDLO_{n+r}F_{n+t}+SDLO_{n+t}F_{n+r} = 2SDFO_{2n+r+t}-\left(-1\right) ^{n+1}L_{r-t}SDFO_{0},$\newline\newline v) $SDFO_{n+r}L_{n+r}+SDFO_{n-r} L_{n-r} = L_{2r}SDFO_{2n}+2\left(-1\right) ^{n+r}SDFO_{0}$.

Proof. The proof of the theorem can be done similarly by the Binet formulas for the $n$th split dual Fibonacci and $n$th split dual Lucas octonions proved above.

4.   Conclusions

In the articles about Quaternions and Octonions in the literature, Catalan's, Cassini's and d'Ocagne's identities have been obtained by using their Binet formulas. In this paper, the Vajda theorems are obtained for split dual Fibonacci and split dual Lucas octonions. We aimed to introduce Vajda theorem, which is not included in the literature and will contribute to finding Catalan's, Cassini's and d'Ocagne's identities. In the Vajda theorem, when we write the criteria in Corollaries 1–3 in our study, Catalan's, Cassini's and d'Ocagne's identities are obtained, respectively. Thus, when we obtain the Vajda theorem by using the Binet formula, we get these three identities, which are well known in the literature, without the need for any other calculations.

Acknowledgments

The authors are grateful to the reviewers' valuable comments that improved the manuscript.

Conflict of interest

The authors declare that they have no conflicts of interest.