We give the representation of Fibonacci quaternions for convenient calculus

http://dx.doi.org/10.5831/HMJ.2016.38.4.675

PROPERTIES OF FUNCTIONS WITH VALUES IN FIBONACCI QUATERNIONS IN CLIFFORD ANALYSIS

Ji Eun Kim and Kwang Ho Shon^∗

Abstract. We give the representation of Fibonacci quaternions for convenient calculus. We research the corresponding properties of Fibonacci quaternions and some examples of a function with values in modified Fibonacci quaternions.

1. Introduction

Quaternion is a member of noncommutative algebra which is ex- tended by the complex numbers. Quaternions are first invented by William Rowan Hamilton in 1843. The algebra of quaternions is de- noted by H. Quaternions are defined in the form

q = q₀+ q₁i + q₂j + q₃k,

where q₀, q₁, q₂ and q₃ are real numbers and i, j and k are standard orthonormal basis in R³which satisfy the quaternion multiplication rules as:

i²= j² = k² = −1,

ij = −ji = k, jk = −kj = i, ki = −ik = j.

Kajiwara et al. [7, 8] studied regenerations in complex, quaternion and Clifford analysis. They extended function spaces to infinite dimensional complex analysis and developed its applications in Clifford analysis. We [9, 10] obtained some results for the regularity of functions on the ternary quaternion and reduced quaternion field in Clifford analysis, and we [11, 12] researched corresponding Cauchy-Riemann systems and proper- ties of a regularity of functions with values in special quaternions such as reduced quaternion, split quaternion, and dual quaternion on Clifford

Received February 3, 2016. Accepted October 14, 2016.

2010 Mathematics Subject Classification. 32A99, 32W50, 30G35, 11E88.

Key words and phrases. Fibonacci number, quaternion, conjugation, Clifford analysis.

∗Corresponding author.

analysis. We [13, 14] investigated the relations between the correspond- ing Cauchy-Riemann system and the corresponding differential opera- tors of special quaternion number systems.

Recently, a new approach is focussed on the extension of Fibonacci numbers into the complex plane. Defining such numbers has been con- sidered by Horadam [5] and Berzsenyi [2]. Horadam defined and de- scribed the generalized Fibonacci sequences and Lucas sequences. Swamy [15] gave the relations of generalized Fibonacci quaternions. Iyer [6]

studied Fibonacci quaternions and obtained some other relations about Fibonacci and Lucas quaternions. Halc [4] expressed the generating function and Binet formulas for these quaternions. Aky˘igit et al. [1]

defined the split Fibonacci and split Lucas quaternions. They also gave Binet formulas and Cassini identities for these quaternions. Moreover, Nurkan et al. [3] defined the dual Fibonacci quaternion and the dual Lu- cas quaternion by combining Fibonacci quaternions, Lucas quaternions and dual quaternions.

In this paper, we give the representation of modified Fibonacci quater- nions. We research the relations between the Fibonacci quaternion and the modified Fibonacci quaternions which connect the Fibonacci num- bers. Also, we give functions defined on the modified Fibonacci quater- nions and investigate expressions of exponential and trigonometric func- tions in Fibonacci quaternions.

2. The representation of modified Fibonacci quaternions We consider modified Fibonacci quaternions with quaterions algebra as follows:

Z = µ_n+ µ_n+1i + µ_n+2j + µ_n+3k,

where µ_n+r (r = 0, 1, 2, 3) are Fibonacci numbers with n ≥ 0. Let z_n+2= µ_n+2+ µ_n+2j + µ_n+2k

and

z_n+1 = µ_n+1+ µ_n+1j + µ_n+1k (n ≥ 0).

Then we give Z = z_n+1+ z_nk with n ≥ 1. The set of modified Fibonacci quaternions is

M_n= {Z = z_n+1+ z_nk | z_n+1, z_n∈ T, n ≥ 1},

which is identical with C², where T = {z | z = x + xj + xk, x ∈ R} is a set of reduced quaternions which is identical with R³.

Proposition 2.1. For any Z ∈ M_n, we have the following properties:

z_nk = k ez_n and zb_nk = kz_n, where

z_n= µ_n+ µ_nj + µ_nk, z_n= µ_n− µ_nj − µ_nk, e

z_n= µ_n− µ_nj + µ_nk, bz_n= µ_n+ µ_nj − µ_nk.

Moreover,

z_nz_n= z_nz_n= ez_nzb_n= bz_nze_n= 3µ²_n.

Proof. Since standard orthonormal bases in R³ are noncommutative, we have

z_nk = (µ_n+ µ_nj + µ_nk)k = k(µ_n− µ_nj + µ_nk) = k ez_n and

b

z_nk = (µ_n+ µ_nj − µ_nk)k = k(µ_n− µ_nj − µ_nk) = kz_n. Moreover, we have

z_nz_n= (µ_n+ µ_nj + µ_nk)(µ_n− µ_nj − µ_nk) = µ²_n+ µ²_n+ µ²_n and

b

z_nze_n= (µ_n+ µ_nj − µ_nk)(µ_n− µ_nj + µ_nk) = µ²_n+ µ²_n+ µ²_n.

Now, we give the addition and the multiplication of M_n: for Z = z_n+1+z_nk and W = w_n+1+w_nk, where w_n+1= z_n+r, w_n= z_n+r−1(n, r ≥ 1),

Z + W = (z_n+1+ w_n+1) + (z_n+ w_n)k and

ZW = (z_n+1w_n+1− z_nwf_n) + (z_n+1w_n+ z_nw]_n+1)k.

From the rule of the multiplication of M_n, we give the conjugate element and the norm of Z in M_n:

Z^∗ = z_n+1− bz_nk,

|Z|²= ZZ^∗ = Z^∗Z = z_n+1z_n+1+ z_nz_n= 3(µ²_n+1+ µ²_n), respectively. Then we have the inverse element of Z ∈ M_n:

Z⁻¹= Z^∗

|Z|².

3. Some examples of functions with values in Fibonacci vari- ables

We introduce the exponential function and the concept of trigono- metric functions and their properties.

Definition 3.1. Let Ω be an open set in M_n. A function exp(Z) is said to be a Fibonacci exponential function on Ω if it satisfies

exp : Ω → M_n, which is given by the convergent series

exp(Z) = X∞ k=0

Z^k k!.

Since Fibonacci quaternions commute with all other quaternions, for µ_n+1, µ_n ∈ R, we have

exp(a + Z) = exp(a) exp(Z) (∀Z ∈ M_n, a ∈ R);

so, if

Z = z_n+1+ z_nk = (µ_n+1+ µ_n+1j + µ_n+1k) + (−µ_n+ µ_ni + µ_nk), then we have

exp(Z) = exp(µ_n+1+ v_n+1) exp(−µ_n+ v_n),

where v_n+1 = µ_n+1j + µ_n+1k and v_n = µ_ni + µ_nk are imaginary (or vector) Fibonacci quaternions.

Theorem 3.1. For Z ∈ M_n, if v_n+1= µ_n+1j +µ_n+1k and v_n= µ_ni+

µ_nk ∈ M_n are imaginary Fibonacci quaternions, putting θ_n+1 = |v_n+1| and θ_n= |v_n|, then we have:

exp(v_n+1) = cos θ_n+1+v_n+1

θ_n+1sin θ_n+1 , exp(v_n) = cos θ_n+v_n θ_nsin θ_n. Furthermore, we can express the exponential function to Fibonacci vari- ables as follows:

exp(Z) = exp(µ_n+1+ v_n+1) exp(−µ_n+ v_n)

= exp(µ_n+1

µ_n )(cos θ_n+1+v_n+1

θ_n+1sin θ_n+1)(cos θ_n+v_n

θ_nsin θ_n).

Proof. We note that:

v_n+1² = (µ_n+1j + µ_n+1k)(µ_n+1j + µ_n+1k) = −µ²_n+1− µ²_n+1= −|v_n+1|² and

v²_n= (µ_ni + µ_nk)(µ_ni + µ_nk) = −µ²_n− µ²_n= −|v_n|²; so,

v²_n+r = −θ_n+r² , v³_n+r = −θ_n+r² v_n+r , v_n+r⁴ = θ_n+r⁴ , v⁵_n+r = θ⁴_n+rv_n+r , v⁶_n+r= −θ_n+r⁶ , · · · , (r = 0, 1) and the series become

exp(v_n+r) = X∞ k=0

v_n+r^k k!

= 1 +v_n+r

1! −θ²_n+r

2! −θ_n+r² v_n+r

3! +θ⁴_n+r

4! + θ_n+r⁴ v_n+r

5! − θ_n+r⁶ 6! + · · ·

= 1 +v_n+r

1! −θ²_n+r

2! −θ³v_n+r

3!θ_n+r +θ⁴_n+r

4! +θ_n+r⁵ v_n+r

5!θ_n+r −θ_n+r⁶ 6! + · · ·

=

³

1 −θ²_n+r

2! + θ_n+r⁴

4! −θ⁶_n+r 6! + · · ·

´

+v_n+r θ_n+r

³ v_n+r

1! −θ_n+r³

3! +θ⁵_n+r 5! + · · ·

´

= cos θ_n+r+v_n+r

θ_n+r sin θ_n+r (r = 0, 1).

So, the exponential function of Fibonacci quaternions is:

exp(Z) = exp(µ_n+1

µ_n )(cos θ_n+1+v_n+1

θ_n+1sin θ_n+1)(cos θ_n+v_n

θ_nsin θ_n).

With the help of the exponential function, Fibonacci quaternionic analogues of the trigonometric functions can be introduced as follows:

Theorem 3.2. For Z ∈ M_n, we can express the trigonometric func- tion to Fibonacci variables as follows:

cos(Z) = cos(µ_n+1− µ_n)H₁+v_n

θ_nsin(µ_n+1− µ_n)H₂ and

sin(Z) = sin(µ_n+1− µ_n)H₁+v_n

θ_ncos(µ_n+1− µ_n)H₂,

where

H₁ = cosh(µ_n+1) cosh(µ_n) −v_n+1v_n

θ_n+1θ_nsinh(µ_n+1) sinh(µ_n) and

H₂= cosh(µ_n+1) sinh(µ_n) − v_n+1v_n

θ_n+1θ_nsinh(µ_n+1) cosh(µ_n).

Proof. By definitions and properties of trigonometric functions to real and complex variables, we have

cos(Z) = cos(µ_n+1+ v_n+1) cos(−µ_n+ v_n)

− sin(µ_n+1+ v_n+1) sin(−µ_n+ v_n) and

sin(Z) = sin(µ_n+1+ v_n+1) cos(−µ_n+ v_n) + cos(µ_n+1+ v_n+1) sin(−µ_n+ v_n).

Since we have the following equations

cos(µ_n+1+ v_n+1) = cos(µ_n+1) cos(v_n+1) − sin(µ_n+1) sin(v_n+1)

= cos(µ_n+1) cosh(µ_n+1) −v_n+1

θ_n+1sin(µ_n+1) sinh(µ_n+1), cos(−µ_n+ v_n) = cos(µ_n) cos(v_n) + sin(µ_n) sin(v_n)

= cos(µ_n) cosh(µ_n) +v_n

θ_nsin(µ_n) sinh(µ_n), sin(µ_n+1+ v_n+1) = sin(µ_n+1) cos(v_n+1) + cos(µ_n+1) sin(v_n+1)

= sin(µ_n+1) cosh(µ_n+1) +v_n+1

θ_n+1cos(µ_n+1) sinh(µ_n+1), and

sin(−µ_n+ v_n) = cos(µ_n) sin(v_n) − sin(µ_n) cos(v_n)

= − sin(µ_n) cosh(µ_n) +v_n

θ_ncos(µ_n) sinh(µ_n),

we have

cos(Z) = cos(µ_n+1+ v_n+1) cos(−µ_n+ v_n)

− sin(µ_n+1+ v_n+1) sin(−µ_n+ v_n)

=

³

cos(µ_n+1) cosh(µ_n+1) −v_n+1

θ_n+1sin(µ_n+1) sinh(µ_n+1)

´

³

cos(µ_n) cosh(µ_n) +v_n

θ_nsin(µ_n) sinh(µ_n)

´

−

³

sin(µ_n+1) cosh(µ_n+1) +v_n+1

θ_n+1cos(µ_n+1) sinh(µ_n+1)

´

³

− sin(µ_n) cosh(µ_n) + v_n

θ_n cos(µ_n) sinh(µ_n)

´

= cos(µ_n+1− µ_n)

³

cosh(µ_n+1) cosh(µ_n) −v_n+1v_n

θ_n+1θ_nsinh(µ_n+1) sinh(µ_n)

´

+v_n

θ_nsin(µ_n+1− µ_n)

³

cosh(µ_n+1) sinh(µ_n) − v_n+1v_n

θ_n+1θ_nsinh(µ_n+1) cosh(µ_n)

´ . Similarly, we also obtain

sin(Z) = sin(µ_n+1+ v_n+1) cos(−µ_n+ v_n) + cos(µ_n+1+ v_n+1) sin(−µ_n+ v_n)

=

³

sin(µ_n+1) cosh(µ_n+1) +v_n+1

θ_n+1cos(µ_n+1) sinh(µ_n+1)

´

³

cos(µ_n) cosh(µ_n) +v_n

θ_nsin(µ_n) sinh(µ_n)

´

+

³

cos(µ_n+1) cosh(µ_n+1) −v_n+1

θ_n+1sin(µ_n+1) sinh(µ_n+1)

´

³

− sin(µ_n) cosh(µ_n) +v_n

θ_ncos(µ_n) sinh(µ_n)

´

= sin(µ_n+1− µ_n)

³

cosh(µ_n+1) cosh(µ_n) −v_n+1v_n

θ_n+1θ_nsinh(µ_n+1) sinh(µ_n)

´

+v_n

θ_ncos(µ_n+1− µ_n)

³

cosh(µ_n+1) sinh(µ_n) − v_n+1v_n

θ_n+1θ_nsinh(µ_n+1) cosh(µ_n)

´ .

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Ji Eun Kim

Department of Mathematics, Pusan National University, Busan 46241, Korea.

E-mail: [email protected] Kwang Ho Shon

Department of Mathematics, Pusan National University, Busan 46241, Korea.

E-mail: [email protected]