View of ON SPLIT HYPERBOLIC K-FIBONACCI QUATERNIONS

ON SPLIT HYPERBOLIC K-FIBONACCI QUATERNIONS

1

Dr. Rajesh Vyas,

²

Dr. Manjeet Singh Teeth and

³

Ms. Neelam Dawar

1

Department of Mathematics Christian Eminent College, Indore (M.P.), India

2

Department of Mathematics Christian Eminent College, Indore (M.P.), India

3

School of Mathematics DAVV Indore (M.P.), India

Abstract- In this paper, we introduce the Split hyperbolic k-Fibonacci quaternions.

Also, some algebraic properties of Split hyperbolic k-Fibonacci quaternions which are connected with hyperbolic numbers and k- Fibonacci numbers are investigated.

Keywords: Fibonacci number, k-Fibonacci numbers, k-Fibonacci quaternions, hyperbolic k-Fibonacci quaternions, Split hyperbolic k-Fibonacci quaternions.

1. INTRODUCTION

First described by Irish Mathematician Hamilton in 1843. Hamilton [10] introduced a set of quaternions which can be represented as

Where

Now let’s talk about the work done on Fibonacci quaternions. Horadam [11]

introduced complex Fibonacci and Lucas quaternions which can be represented by

And

Where

In Ramirej [12] has defined the nth k-Fibonacci and nth k-Lucas quaternions as

And

Where i , j , k satisfy the multiplication rule (2).

In 2013, Halici [5], defined complex Fibonacci quaternions as

Where i, j, k are hyperbolic quaternions units which satisfy the multiplication rules

In 2015, polath Kizilates and kesim [2] defined Split k-Fibonacci and Split k-Lucas quaternions and respectively as follows :

and

Where i, j, k are Splits quaternions units which satisfy the multiplication rules

In 2018, F gen Torunbalci Aydin [1] defined Hyperbolic k-Fibonacci quaternions and the hyperbolic k-Lucas quaternions respectively as follows:

and

Where

In this paper, the Split hyperbolic k-Fibonacci quaternions and the Split hyperbolic k-Lucas quaternions will be defined respectively as follows

and

Inspired by these, In this paper, we introduce the Split hyperbolic k-Fibonacci quaternions and hyperbolic k- Lucas quaternion. We give some properties for the Split hyperbolic k-Fibonacci and hyperbolic k-Lucas quaternions.

2. SPLIT HYPERBOLIC K-FIBONACCI QUATERNIONS

The k-Fibonacci sequence

is defined as

Here k is positive real number. In this section, firstly Split hyperbolic k-Fibonacci quaternions will be defined. Split hyperbolic k-Fibonacci quaternion are defined by using the k- Fibonacci numbers hyperbolic quaternion units as follows

Where

be two Split hyperbolic k-Fibonacci quaternions such that

and

Then, the addition and substraction of two Split hyperbolic k- Fibonacci quaternions are defined by,

Multiplication of two Split hyperbolic k-Fibonacci quaternion is defined by

The scaler and the vector parts of Split hyperbolic k-Fbonacci quaternion are denoted by and

Thus, The conjugate of Split hyperbolic k-

Fibonacci quaternions is denoted by and it is

quaternions are given.

Theorem 2.1. Let be the n-th terms of k-Fibonacci sequence and Split hyperbolic k-

Fibonacci quaternions respectively. In this case, for we can give the following relations:

Proof .(1): By equation (2.2) we get,

(2): By equation (2.2) we get,

(3): By equation (2.2) we get,

Theorem2.2. Let be conjugation of Split hyperbolic k-Fibonacci quaternions In this case, we can give the following relations between these quaternions:

Proof. (1): By using (2.2) and (2.6), we get,

(2): By using (2.2) and (2.6), we get,

(3): By using (2.2) and (2.6), we get,

Where the identity of k-Fibonacci number Ramirez [12]

was used.

(4): By using theorem [2.2] relation 2. we get,

Here the Honsberger identity of k-Fibonacci number in

Falcon and plaza [14] was used.

Here the identity of k-Fibonacci number [13] was used.

3.CONCLUSION

In this study, a number of new results on Split hyperbolic k-Fibonacci quaternions were derived. Quaternion have great importance in mathematics.

REFERENCES

1. Aydin, F.T., Hyperbolic k-Fibonacci quaternions, arXiv:1812.00781v1, 2018.

2. E. Polath, C. Kizilates and S. Kesim, On split k-Fibonacci and k-Lucas quaternions, Advances in Applied Clifford Algebras, 26(1) (2016) 353-362.

3. P. Catarino, On Some Identities for k-Fibonacci sequence, Int.J. Contemp . Math. Sciences, Vol.9,(2014), 37-42.

4. Kosal, I. A., A note on Hyperbolic Quaternions, Universal Journal of Mathematics and Applications 1 (2018), 155-159.

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Analysis, Vol. 7, (2013),no. 38, 1877-1884.

10. W. R. Hamilton, Elements of Quaternions, Longmans, Green and Co. London, 1886.

11. A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, The American Mathematical Monthly 70 (3) (1963) 289-291.

12. J. L. Ramirez, Some combinatorial properties of the k-Fibonacci and k-Lucas quaternions, Analele Universitatii “Ovidius” Constanta-Seria Mathematica. 23(2) (2015)201-212.

13. S. Falcon, A. Plaza, On the Fibonacci k-numbers, Choas Solitons Fractals, 32 (2007), no. 5, 1615-1624.

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15. M.R. Iyer, A Note On Fibonacci Quaternions, The Fibonacci Quaterly 7(3) (1969) 225-229.

16. M.R. Iyer, Some Results On Fibonacci Quaternions, The Fibonacci Quaterly 7 (1969) 201-210.