A new generalization of Fibonacci hybrid and Lucas hybrid numbers

Chaos, Solitons and Fractals 130 (2020) 109449

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

A new generalization of Fibonacci hybrid and Lucas hybrid numbers Can Kızılates¸ Department of Mathematics, Art and Science Faculty, Zonguldak Bülent Ecevit University, Zonguldak 67100, Turkey

a r t i c l e

i n f o

Article history: Received 7 May 2019 Revised 24 August 2019 Accepted 13 September 2019

a b s t r a c t In the paper, we define the q−Fibonacci hybrid numbers and the q−Lucas hybrid numbers, respectively. Then, we give some algebraic properties of q−Fibonacci hybrid numbers and the q−Lucas hybrid numbers. © 2019 Elsevier Ltd. All rights reserved.

MSC: 11B39 11R52 05A15 Keywords: Hybrid number Horadam number q−integer Binet-Like formula Exponential generating function

1. Introduction

Fn = Fn−1 + Fn−2 ,

n≥2

W.R.Hamilton defined the four dimensional quaternion algebra in 1843 [1]. Quaternions can be considered as an extension to the complex number and they have been studied in many areas such as computer sciences, physics, differential geometry, quantum physics by the researchers (see for details [2,3]). The real quaternion algebra

Ln = Ln−1 + Ln−2 ,

n≥2

H = {q = q 0 + q 1 i + q 2 j + q 3 k : q s ∈ R , s = 0 , 1 , 2 , 3 } is a 4-dimensional R-vector space with basis {1 ࣃ e0 , i ࣃ e1 , j ࣃ e2 , k ࣃ e3 } satisfying multiplication rules listed in the following table (Table 1). Horadam [4] defined the quaternions with the classic Fibonacci and Lucas number components as

QFn = Fn + Fn+1 e1 + Fn+2 e2 + Fn+3 e3 , and

QLn = Ln + Ln+1 e1 + Ln+2 e2 + Ln+3 e3 , respectively, where Fn and Ln are the n-th classic Fibonacci and Lucas numbers defined by the

E-mail address: [email protected] https://doi.org/10.1016/j.chaos.2019.109449 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

with the initial conditions F0 = 0, F1 = 1, L0 = 2 and L = 1, respectively. Several generalizations of the well-known quaternion numbers as the Fibonacci and Lucas quaternions, Pell and PellLucas quaternions, Jacobsthal and Jacobsthal-Lucas quaternions have been studied by several researchers (see, for example, [5–11]). In [12] Akkus and Kizilaslan defined a more general quaternion sequence that generalized the above papers. On the other hand, Özdemir [13] introduced the set of hybrid numbers denoted by K and contains complex, dual and hyberbolic numbers. The set of hybrid numbers



K = a + bi + c + dh : a, b, c, d ∈ R, i2 = −1,



2 = 0,

h2 = 1, ih = hi =  + i . Let Z1 = a1 + b1 i + c1  + d1 h and Z2 = a2 + b2 i + c2  + d2 h be any two hybrid numbers. The equality, addition, substraction and multiplication by scalar are defined as follows: Z1 = Z2 only if a1 = a2 , b1 = b2 , c1 = c2 , d1 = d2 (Equality), Z1 + Z2 = (a1 + a2 ) + (b1 + b2 )i+(c1 + c2 ) + (d1 + d2 )h (addition), Z1 − Z2 = (a1 − a2 ) + (b1 − b2 )i+(c1 − c2 ) + (d1 − d2 )h (substraction), sZ1 = sa1 + sb1 i + sc1  + sd1 h (multiplication by scalar s ∈ R).

2

C. Kızılates¸ / Chaos, Solitons and Fractals 130 (2020) 109449 Table 1 The multiplication table for the basis of H. ×

e0

e1

e2

e3

e0 e1 e2 e3

e0 e1 e2 e3

e1 −e0 −e3 e2

e2 e3 −e0 −e1

e3 −e2 e1 −e0

In the light of the above-cited recent works, we define two families of the hybrid numbers with components including quantum integers. Thus, our definitions give rise to a more general hybrid number sequence by receiving components from q−integer. These two families of the hybrid numbers are called as the q−Fibonacci hybrid numbers and q−Lucas hybrid numbers, respectively. We give the special cases of the q−Fibonacci hybrid numbers and q−Lucas hybrid numbers studied by many researchers previously. We obtain various results for q−Fibonacci hybrid numbers and q−Lucas hybrid numbers included Binet-Like formulas, exponential generating functions, summation formulas, CatalanLike identities, Cassini-Like identities and d’Ocagne-Like identities, respectively.

Table 2 The multiplication table for the basis of K. .

1

i



1 i

1 i

i −1 h+1 − − i







h

h

1−h 0



h

2. q−Fibonacci hybrid numbers and q−Lucas hybrid numbers

h

+i − 1

Addition operation in the hybrid numbers is both commutative and associative. Zero is the null element. With respect to the addition operation, the inverse element of Z is −Z = −a − bi − c − dh. This implies that, (K, + ) is an Abelian group. The multiplication of hybrid numbers is not commutative, but it has the property of associativity. The multiplication of hybrid numbers as above (Table 2): For a, b, p, q ∈ R, the Horadam numbers Wn = Wn (a, b; p, q ) are defined by the following recurrence relation

Wn = pWn−1 + qWn−2 ,

n≥2

Wn (0, 1; 1, 1 ) = Fn , the Fibonacci sequence, Wn (2, 1; 1, 1 ) = Ln , the Lucas sequence, Wn (0, 1; 2, 1 ) = Pn , the Pell sequence, Wn (2, 2; 2, 1 ) = P Ln , the Pell-Lucas sequence, Wn (0, 1; k, 1 ) = Fk,n , the k−Fibonacci sequence, Wn (2, 1; k, 1 ) = Lk,n , the k−Lucas sequence, Wn (0, 1; 1, 2 ) = Jn , the Jacobsthal sequence, Wn (2, 1; 1, 2 ) = jn , the Jacobsthal-Lucas sequence, Wn (0, 1; 2, k ) = Pk,n , the k−Pell sequence, Wn (2, 2; 2, k ) = P Lk,n , the k−Pell-Lucas sequence.

+ α n+2 [n + 3]q h,

(2.1)

HLn (α ; q ) =

αn

[2n]q [n]q

+ α n+1

[2n + 2]q

[2n + 6]q [n + 3]q

[n + 1]q h.

i+α n+2

[2n + 4]q [n + 2]q

 (2.2)

√

Aα n − Bβ n , α−β

1 − qn = 1 + q + q2 + . . . + qn−1 . 1−q

α n−1 [n]q + α n [n + 1]q i+α n+1 [n + 2]q 

Some special cases of q−hybrid Fibonacci numbers HFn (α ; q ) and q−hybrid Lucas numbers HLn (α ; q ) are as follows: 1. For α = 1+2 5 and q = −1 , the q−hybrid Fibonacci numbers α2 HFn (α ; q ) become the Fibonacci hybrid numbers HFn , √ 2. For α = 1+2 5 and q = −1 , the q−Lucas hybrid numbers α2 HLn (α ; q ) become the Lucas hybrid numbers HLn , √ 3. For α = 1 + 2 and q = −1 , the q− Fibonacci hybrid numα2 bers HFn (α ; √ q ) become the Pell hybrid numbers HPn , 4. For α = 1 + 2 and q = −1 , the q−Lucas hybrid numbers α2 HLn (α ; q ) become the Pell-Lucas hybrid numbers HP Ln , √ 2

where A = b − aβ , B = b − aα , α and β are the roots of the x2 − px − q = 0. Now, we give some notations related to q−calculus. q−calculus may be viewed as a generalization of ordinary calculus. Thus, several mathematical concepts are generalized thanks to q−calculus. The theory of the quantum (q−) calculus has been extensively studied in many branches of mathematics as well as in other areas in biology, physics, electrochemistry, economics, probability theory, and statistics. For n ∈ N0 , we give the q−integer [n]q

[n]q =

HFn (α ; q ) =

+ α n+3

It is well known that for n ≥ 0, Horadam numbers can be expressed by using the following Binet-Like formula:

Wn =

Definition 1. The q−hybrid Fibonacci numbers and q−hybrid Lucas numbers are defined by

and

with the initial values W0 = a and W1 = b [14,15]. Some special cases of the Horadam sequence Wn (a, b; p, q) are as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

In this section, we define a new generalization of Fibonacci hybrid numbers and Lucas hybrid numbers called q−Fibonacci hybrid numbers and q− Lucas hybrid numbers, respectively. We give Binet-Like formulas, exponential generating functions and some other identities for the q−Fibonacci hybrid numbers and the q−Lucas hybrid numbers. In every part of the paper, we take n ∈ N and 1 − q as a nonzero number.

(1.1)

By virtue of (1.1), for all m, n ∈ Z, we can easily find that

[m + n]q = [m]q + qm [n]q . For more details related to the quantum (q−) calculus, we refer to [16,17].

5. For α = k+ 2k +4 and q = −1 , the q−Fibonacci hybrid numα2 bers HFn (α ; q ) become the k−Fibonacci hybrid numbers HFk,n , √ 2 6. For α = k+ 2k +4 and q = −1 , the q−Lucas hybrid numbers α2 HLn (α ; q ) become the k−Lucas hybrid numbers HLk,n , 7. For α = 2 and q = −1 2 , the q−Fibonacci hybrid numbers HFn (α ; q ) become the Jacobsthal hybrid numbers HJn , 8. For α = 2 and q = −1 2 , the q−Lucas hybrid numbers HLn (α ; q ) become the Jacobsthal-Lucas hybrid numbers H jn , √ 9. For α = 1 + 1 + k and q = −k , the q−Fibonacci hybrid α2 numbers HFn (α ; q ) become the k−Pell hybrid numbers HPk,n , √ 10. For α = 1 + 1 + k and q = −k , the q−Lucas hybrid numbers α2 HLn (α ; q ) become the k−Pell-Lucas hybrid numbers HP Lk,n . In the literature these special cases have been studied by many researchers (for details, [18–22], see also the closely-related earlier works cited therein).

C. Kızılates¸ / Chaos, Solitons and Fractals 130 (2020) 109449

3

Theorem 2. The Binet-Like formulas for the q−Fibonacci hybrid numbers HFn (α ; q ) and q−Lucas hybrid numbers HLn (α ; q ) are

α n γˆ − (α q )n δ , α (1 − q )

(2.3)

HLn (α ; q ) = α n γˆ + (α q )n δ,

(2.4)

HFn (α ; q ) = and

where γˆ = 1 + α i + α 2  + α 3 h and  δ = 1 + ( α q )i + ( α q )2  + ( α q )3 h. Proof. By virtue of (2.1) and (1.1), we find that

HFn (α ; q ) = α n−1 [n]q + α n [n + 1]q i+α n+1 [n + 2]q  + α n+2 [n + 3]q h 1 − qn 1 − qn+1 1 − qn+2 1 − qn+3 + α n+1 i + α n+2  + α n+3 h α − αq α − αq α − αq α − αq α n γˆ − (α q )n δ = . α (1 − q ) = αn

In a similar way, equality (2.4) can be derived.



Theorem 3. The exponential generating functions for the q−Fibonacci hybrid numbers and q−Lucas hybrid numbers are ∞ 

HFn (α ; q )

tn γˆ eαt −  δ e(αq)t = , n! α (1 − q )

(2.5)

HLn (α ; q )

tn = γˆ eαt +  δ e(αq)t . n!

(2.6)

n=0

and ∞  n=0

Proof. By virtue of Binet formula for the q−Fibonacci hybrid numbers, we have ∞ 

HFn (α ; q )

n=0

∞  tn = n!



n=0

=

 δ tn α n γˆ − (α q )n α (1 − q ) n!

∞  (αt )n − α (1 − q ) n!

γˆ

n=0

=

γˆ eαt −  δ e(αq)t α (1 − q )

 δ

∞  (α qt )n α (1 − q ) n! n=0

.

The result (2.6) can be similarly obtained.



Theorem 4. For nonnegative integer numbers n and k, we have

  

n  n i i=0

  

n  n i i=0

−α q 2

−α 2 q

n−i n−i



HF2i+k (α ; q ) =

HL2i+k (α ; q ) =

 2 HFn+k (α ; q ) n

if n is even,

 2 HLn+k (α ; q )

n  2 HLn+k (α ; q )

if n is even,

HFn+k (α ; q )

if n is odd,

n−1



n+1 2

if n is odd,

(2.8)

 

n   n−i  n n (−1 )i −α 2 q HF2i+k (α ; q ) = −α [2]q HFn+k (α ; q ), i i=0

(2.7)

(2.9)

 

n   n−i  n n (−1 )i −α 2 q HL2i+k (α ; q ) = −α [2]q HLn+k (α ; q ). i

(2.10)

i=0

where  = (α − α q )2 . Proof. We prove the equality (2.7). The results (2.8), (2.9) and (2.10) can be similarly obtained. By virtue of Binet-Like formula (2.3), we have

  

n  n i i=0

−α q 2

n−i

 



n−i α 2i+k γˆ − (α q )2i+k δ α ( 1 − q) i=0  n  n 1 = δ α 2 − α 2 q α k γˆ − α 2 q2 − α 2 q (α q )k α − αq  √ n k √ δ α  α γˆ − (−α q )n (α q )k = . α − αq

n  n HF2i+k (α ; q ) = i

−α 2 q

(2.11)

4

C. Kızılates¸ / Chaos, Solitons and Fractals 130 (2020) 109449

If n is even in (2.11), we get

  

n  n i

−α 2 q

n−i

HF2i+k (α ; q ) =

i=0

 √ n k √ δ α  α γˆ − (α q )n (α q )k α − αq

α n+k γˆ − (α q )n+k δ α (1 − q ) n =  2 HFn+k (α ; q ). =

√



n

If n is odd in (2.11), we get

  

n  n i

−α q 2

n−i

i=0

 √ n k √ δ α  α γˆ + (α q )n (α q )k HF2i+k (α ; q ) = α − αq =

√



=

n−1

n−1 2

(α n+k γˆ + (α q )n+k δ)

HLn+k (α ; q ).



So the proof is completed.

Theorem 5. For nonnegative integer number n, we have



1 n  2 + α n [2]nq HFn (α ; q ) if n is even, 2 −α q HF4i (α ; q ) = 1  n−1 n n 2i 2  HLn (α ; q ) − α [2]q HFn (α ; q ) if n is odd, 2 i=0  

1 n n n n  n  n−i  2 + α [2]q HLn (α ; q ) if n is even, 2 −α 2 q HL4i (α ; q ) = 1  n+1 n n 2i  2 HFn − α [2]q HLn (α ; q ) if n is odd, 2 i=0   

n 1 n   2 n−i  2 − α n [2]nq HFn−1 (α ; q ) if n is even, n 2 −α q HF4i+1 (α ; q ) = 1  n−1 n n 2i + 1 2 HL  ( α ; q ) + α 2 HF ( α ; q ) if n is odd, [ ] n−1 n−1 q 2 i=0   

n n 1   2 n−i  2 − α n [2]nq HLn−1 (α ; q ) if n is even, n −α q HL4i+1 (α ; q ) = 1  2n+1 2i + 1  2 HFn−1 + α n [2]nq HLn−1 (α ; q ) if n is odd, 2 i=0 n 

  n 

2

n−i

(2.12)

(2.13)

(2.14)

(2.15)

Proof. Firstly we prove the equality (2.12). By virtue of Theorem (4), we find that n  i=0

  n  2i

−α 2 q

n−i

 

HF4i (α ; q ) =

n  n−i 1 n (1 + (−1 )i ) −α 2 q HF2i (α ; q ) 2 i i=0

 

n 1  n = 2 i



−α 2 q

i=0

n−i

 

n   n−i n HF2i (α ; q ) + (−1 )i −α 2 q HF2i (α ; q ) i



i=0



1 n  2 + α n [2]nq HFn (α ; q ) 2 = 1  n−1  2 HLn (α ; q ) − α n [2]nq HFn (α ; q ) 2

if n is even, if n is odd,

.

So the proof is completed. Equalities (2.13), (2.14) and (2.15) can be similarly obtained.



Theorem 6. (Catalan-Like Identity). Let n and r be arbitrary positive integers such that n ≥ r. Then we have

α 2n−2 qn (1 − qr )( δ γˆ − q−r γˆ  δ) , ( 1 − q )2  HLn+r (α ; q )HLn−r (α ; q ) − HL2n (α ; q ) = α 2n qn−r (1 − qr ) γˆ  δ − qr  δ γˆ . HFn+r (α ; q )HFn−r (α ; q ) − HF2n (α ; q ) =

(2.16) (2.17)

Proof. By using the Binet formula of the q−Fibonacci hybrid numbers, we have the left hand-side of the equality (2.16),



HFn+r (α ; q )HFn−r (α ; q ) −

HF2n

 n−r   n 2 δ δ δ α n+r γˆ − (α q )n+r  α γˆ − (α q )n−r  α γˆ − (α q )n (α ; q ) = − . α (1 − q ) α (1 − q ) α (1 − q )

After some elementary calculations, we get

HFn+r (α ; q )HFn−r (α ; q ) − HF2n (α ; q ) =

α 2n−2 qn (1 − qr )( δ γˆ − q−r γˆ  δ) . ( 1 − q )2

The result (2.17) can be similarly obtained.



Theorem 7. (Cassini-Like Identity). For n ≥ 1, the following equalities hold:

α 2n−2 qn ( δ γˆ − q−1 γˆ  δ) , (1 − q )  HLn+1 (α ; q )HLn−1 (α ; q ) − HL2n (α ; q ) = α 2n qn−1 (1 − q ) γˆ  δ − q δ γˆ . HFn+1 (α ; q )HFn−1 (α ; q ) − HF2n (α ; q ) =

(2.18) (2.19)

C. Kızılates¸ / Chaos, Solitons and Fractals 130 (2020) 109449

Proof. Since the Cassini-Like identity is a special case for r = 1 of Catalan-Like identity, the proof is trivial. 

5

Declaration of Competing Interest

Theorem 8. (d’Ocagne-Like Identity). Let n be a nonnegative integer and m a natural number. If m > n + 1, then we have

The author declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

HFm (α ; q )HFn+1 (α ; q ) − HFm+1 (α ; q )HFn (α ; q ) α m+n−1 (qn γˆ  δ − qm  δ γˆ ) = , (1 − q )

Acknowledgments

HLm (α ; q )HLn+1 (α ; q ) − HLm+1 (α ; q )HLn (α ; q ) = α m+n+1 (q − 1 )(qn γˆ  δ − qm  δ γˆ ).

(2.20) The author would like to thank the anonymous reviewer for his/her valuable comments and suggestions to improve the quality of the paper.

(2.21)

Proof. By using the Binet-Like formula of the q−Fibonacci hybrid numbers, we have

HFm (α ; q )HFn+1 (α ; q ) − HFm+1 (α ; q )HFn (α ; q ) α m γˆ − (α q )m δ α n+1 γˆ − (α q )n+1 δ = α (1 − q ) α (1 − q ) α m+1 γˆ − (α q )m+1 δ α n γˆ − (α q )n δ − . α (1 − q ) α (1 − q ) After some calculus, we easily seen that

HFm (α ; q )HFn+1 (α ; q ) − HFm+1 (α ; q )HFn (α ; q ) α m+n−1 (qn γˆ  δ − qm  δ γˆ ) = . (1 − q ) The result (2.21) can be similarly obtained.



3. Conclusion In our present investigation, we have introduced and studied systematically q−Fibonacci hybrid numbers and q−Lucas hybrid numbers which are defined by means of the q−integer. We have derived several interesting properties of q−Fibonacci hybrid numbers and q−Lucas hybrid numbers such as Binet-Like formulas, exponential generating functions, summation formulas, Cassini-Like identities, Catalan-Like identities and d’Ocagne-Like identities. We show that the new hybrid numbers that we introduce include previously introduced Fibonacci, Lucas, Pell, Pell–Lucas, Jacobsthal and Jacobsthal–Lucas, k−Pell, k−Pell-Lucas hybrid numbers. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

References [1] Hamilton WR. Elements of Quaternions. London: Longman, Green, & Company; 1866. [2] Baez J. The octonions. Bull Amer Math Soc 2002;39(2):145–205. [3] Hankins TL. Sir William Rowan Hamilton. Baltimore: Johns Hopkins University Press; 1980. [4] Horadam AF. Complex fibonacci numbers and fibonacci quaternions. Am Math Monthly 1963;70:289–91. [5] Horadam AF. Quaternion recurrence relations. Ulam Q 1993;2:22–33. [6] Iyer MR. A note on fibonacci quaternions. Fibonacci Q 1969;3:225–9. [7] Szynal-Liana A, Wloch I. The pell quaternions and the pell octonions. Adv Appl Clifford Algebras 2016;26(1):435–40. [8] Szynal-Liana A, Wloch I. A note on jacobsthal quaternions. Adv Appl Clifford Algebras 2016;26:441–7. [9] Cimen CB, Ipek A. On pell quaternions and pell–lucas quaternions. Adv Appl Clifford Algebras 2016;26(1):39–51. [10] Halici S. On fibonacci quaternions. Adv Appl Clifford Algebras 2012;22(2):321–7. [11] Catarino P. A note on h(x)-fibonacci quaternion polynomials. Chaos Solitons Fractals 2015;77:1–5. [12] Akkus I, Kizilaslan G. Quaternions: quantum calculus approach with applications. Kuwait J Sci 2019. In prees. [13] Özdemir M. Introduction to hybrid numbers. Adv Appl Clifford Algebras 2018;28:11. doi:10.10 07/s0 0 0 06- 018- 0833- 3. [14] Horadam AF. Basic properties of a certain generalized sequence of numbers. The Fibonacci Q 1965;3:161–76. [15] Horadam AF. Generating functions for powers of a certain generalized sequence of numbers. Duke Math J 1965;32:437–46. [16] Kac V, Cheung P. Quantum Calculus Universitext. New York: Springer-Verlag; 2002. [17] Andrews GE, Askey R, Roy R. Special functions. New York: Cambridge Univ. Press; 1999. [18] Szynal-Liana A. The horadam hybrid numbers. Discuss Math Gen Algebra Appl 2018;38(1):91–8. [19] Szynal-Liana A, Wloch I. On jacosthal and jacosthal-lucas hybrid numbers. Ann Math Sil 2018. doi:10.2478/amsil- 2018- 0 0 09. [20] Szynal-Liana A, Wloch I. On pell and pell-lucas hybrid numbers. Commentat Math 2018;58:11–17. [21] Catarino P. On k−pell hybrid numbers. J Discrete Math Sci Cryptography 2019;22(1):83–9. [22] Cerda Moreles G. Investigation of generalized hybrid fibonacci numbers and their properties. 2018. arXiv:1806.02231.