Polatli and Kesim Advances in Difference Equations (2015) 2015:169 DOI 10.1186/s13662-015-0511-x
RESEARCH
Open Access
On quaternions with generalized Fibonacci and Lucas number components Emrah Polatli* and Seyhun Kesim *
Correspondence:
[email protected] Department of Mathematics, Bulent Ecevit University, Zonguldak, 67100, Turkey
Abstract In this paper, we give the exponential generating functions for the generalized Fibonacci and generalized Lucas quaternions, respectively. Moreover, we give some new formulas for binomial sums of these quaternions by using their Binet forms. MSC: 11B39 Keywords: generalized Fibonacci quaternions; generalized Lucas quaternions
1 Introduction The well-known Fibonacci and Lucas sequences are defined by the following recurrence relations: for n ≥ , Fn+ = Fn+ + Fn and Ln+ = Ln+ + Ln , where F = , F = , L = , and L = , respectively. Here, Fn is the nth Fibonacci number and Ln is the nth Lucas number. The Binet formulas for the Fibonacci and Lucas sequences are given by Fn =
αn – β n α–β
and Ln = α n + β n , where √ + α=
and
√ – β= .
The numerous relations connecting Fibonacci and Lucas numbers have been given by Vajda []. © 2015 Polatli and Kesim. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Polatli and Kesim Advances in Difference Equations (2015) 2015:169
Carlitz, Ferns, Layman and Hoggatt studied many properties of binomial sums of these numbers (see [–]). The generalized Fibonacci and Lucas sequences, {Un } and {Vn }, are defined by the following recurrence relations: for n ≥ and any nonzero integer p, Un+ = pUn+ + Un and Vn+ = pVn+ + Vn , where U = , U = , V = , and V = p, respectively. If we take p = , then Un = Fn (nth Fibonacci number) and Vn = Ln (nth Lucas number). Let λ and μ be the roots of the characteristic equation x – px – = . Then the Binet formulas for the sequences {Un } and {Vn } are given by Un =
λn – μn λ–μ
and Vn = λn + μn , where λ = (p + p + )/ and μ = (p – p + )/. √ From these equalities, by putting = p + , we see that + λ = λ and + μ = √ –μ . The generating function and the exponential generating function of a sequence {an } are defined by
a(x) =
∞
an xn
n=
and aˆ (x) =
∞ n=
an
xn , n!
respectively. Now, we would like to explain how the generating function of a given sequence {an } is derived: Firstly, we write down a recurrence relation which is a single equation expressing an in terms of other elements of the sequence. This equation should be valid for all integers n, assuming that a– = a– = · · · = . Then we multiply both sides of the equation by xn and n sum over all n. This gives, on the left, the sum ∞ n= an x , which is the generating function a(x). The right-hand side should be manipulated so that it becomes some other expression involving a(x). Lastly, if we solve the resulting equation, we get a closed form for a(x). For more details and properties related to the generating functions and special functions, we refer to [–].
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A quaternion p with real components a , a , a , a , and basis , i, j, k is an element of the form p = a + a i + a j + a k = (a , a , a , a ) (a = a ), where i = j = k = –, ij = k = –ji,
jk = i = –kj,
ki = j = –ik.
Throughout this paper, we work in the real division quaternion algebra without specific references being given. The nth Fibonacci and the nth Lucas quaternions were defined by Horadam in [] as Qn = Fn + Fn+ i + Fn+ j + Fn+ k and Kn = Ln + Ln+ i + Ln+ j + Ln+ k, respectively, where Fn and Ln are the nth Fibonacci number and the nth Lucas number. Similar to Horadam, the nth generalized Fibonacci quaternion and nth generalized Lucas quaternion are defined by
Qn = Un + Un+ i + Un+ j + Un+ k and
Kn = Vn + Vn+ i + Vn+ j + Vn+ k, respectively []. Here, Un and Vn are the nth generalized Fibonacci number and the nth generalized Lucas number. For n ≥ , the following recurrence relations hold:
Qn+ = pQn+ + Qn and
Kn+ = pKn+ + Kn , where Q = , , p, p + , Q = , p, p + , p + p , K = , p, p + , p + p , K = p, p + , p + p, p + p + .
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In [], Iakin gave the Binet formulas for the generalized Fibonacci and generalized Lucas quaternions as follows:
Qn =
λλn – μμn λ–μ
and
Kn = λλn + μμn , where λ = + λi + λ j + λ k and μ = + μi + μ j + μ k. Iyer [] gave some relations connecting the Fibonacci and Lucas quaternions. In [], Swamy derived the relations of generalized Fibonacci quaternions. Iakin [, ] introduced the concept of a higher order quaternion and generalized quaternions with quaternion components. Horadam [] studied the quaternion recurrence relations. In [], Halici derived generating functions and many identities for Fibonacci and Lucas quaternions. In [], Flaut and Shpakivskyi investigated some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions. Very recently, Ramirez [] has obtained some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. Inspired by these results, in the present paper we give the exponential generating functions for the generalized Fibonacci and generalized Lucas quaternions. Moreover, we derive some new formulas for binomial sums of these quaternions by using their Binet forms.
2 Some properties of generalized Fibonacci and Lucas quaternions Theorem . ([]) The generating functions for the generalized Fibonacci and generalized Lucas quaternions are F(t) =
Q + (Q – pQ )t – pt – t
L(t) =
K + (K – pK )t , – pt – t
and
respectively. Proof Let F(t) = ∞
∞
Qn t n – pt
n=
n= Qn t
∞
n
and L(t) =
Qn t n – t
n=
= Q + (Q – pQ )t +
∞
∞
n= Kn t
n
. Then we get the following equation:
Qn t n
n= ∞
(Qn – pQn– – Qn– )t n .
n=
Since, for each n ≥ , the coefficient of t n is zero in the right-hand side of this equation, we obtain F(t) =
Q + (Q – pQ )t . – pt – t
Polatli and Kesim Advances in Difference Equations (2015) 2015:169
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Similarly, we obtain L(t) =
K + (K – pK )t . – pt – t
Theorem . The exponential generating functions for the generalized Fibonacci and generalized Lucas quaternions are ∞ Qn n=
n!
tn =
λeλt – μeμt λ–μ
and ∞ Kn n=
n!
t n = λeλt + μeμt .
Proof By the Binet formula for the generalized Fibonacci quaternions, we get ∞ Qn n=
n!
n
t =
∞ λλn – μμn t n λ–μ
n=
n!
∞
=
=
∞
μ (μt)n λ (λt)n – λ – μ n= n! λ – μ n= n! λeλt – μeμt
.
λ–μ
Similarly, by the Binet formula for the generalized Lucas quaternions we obtain ∞ Kn n=
n!
tn =
∞
λλn + μμn
tn
n=
=λ
∞ (λt)n n=
n!
+μ
n! ∞ (μt)n n=
n!
= λeλt + μeμt .
Now, we give the following theorems in which the first formulas are proved, and the remaining formulas can be obtained similarly. Theorem . For n, k ≥ , we have n n n if n is even, Qn+k Qi+k = n– i Kn+k if n is odd, i= n n n if n is even, Kn+k Ki+k = n+ i Qn+k if n is odd, i= n n (–)i Qi+k = (–p)n Qn+k , i i= n n (–)i Ki+k = (–p)n Kn+k . i i=
Polatli and Kesim Advances in Difference Equations (2015) 2015:169
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Proof If we use the Binet formula for the generalized Fibonacci quaternions, we get n n n n λλi+k – μμi+k Qi+k = λ–μ i i i= i= n n λλk n i μμk n i = λ – μ λ – μ i= i λ – μ i= i
=
n μμk n λλk + λ – + μ λ–μ λ–μ
√ λλk √ n μμk (λ ) – (–μ )n λ–μ λ–μ n if n is even, Qn+k = n– Kn+k if n is odd. =
Theorem . ([]) For n ≥ , we have n n i p Qi = Qn , i i= n n i p Ki = Kn . i i=
Proof By the Binet formula for the generalized Fibonacci quaternions, we have n n n i n i λλi – μμi p Qi = p λ–μ i i i= i= n n μ λ n n i = (pλ) – (pμ)i λ – μ i= i λ – μ i= i
μ λ ( + pλ)n – ( + pμ)n λ–μ λ–μ n λλ – μμn = λ–μ =
= Qn .
Theorem . For n ≥ , we have n n Qi = i i= n n Ki = i i=
n ( + pn )Qn n– ( Kn – pn Qn )
if n is even, if n is odd,
n ( + pn )Kn n+ ( Qn – pn Kn )
if n is even, if n is odd,
n n Qi+ = i + i= n n Ki+ = i + i=
n ( – pn )Qn n– ( Kn + pn Qn )
if n is even, if n is odd,
n ( – pn )Kn n+ ( Qn + pn Kn )
if n is even, if n is odd.
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Proof By Theorem ., we obtain n n n n + (–)i Qi Qi = i i i= i=
n n n n i (–) Qi = Qi + i= i i i= n ( + pn )Qn if n is even, = n– n ( Kn – p Qn ) if n is odd.
Theorem . For n ≥ , we have n– n n (Kn – (V + p)Vn+ – (V + V )Vn ) if n is even, (Q i ) = n– i (Qn – (V + p)Un+ – (V + V )Un ) if n is odd, i= n n n if n is even, (Kn – (V + p)Vn+ – (V + V )Vn ) (K i ) = n+ (Qn – (V + p)Un+ – (V + V )Un ) i if n is odd. i= Proof From the quaternion multiplication, we have (λ) = λ – (V + p)λ + V + V , (μ) = μ – (V + p)μ + V + V . If we consider the Binet formula for the generalized Fibonacci quaternions, we have n n n n λλi – μμi (Q i ) = i i λ–μ i= i= n
n i (λ) λ + (μ) μi – λμ(λμ)i = i= i n n
λμ n i n i = (λ) λ + (μ) μ – (–)i i= i i= i n n (λ) n i (μ) n i = λ + μ i= i i= i
n (μ) n (λ) + λ + + μ √ √ (μ) (λ) = (λ )n + (–μ )n n– n = (λ) λ + (μ) (–μ)n n– (Kn – (V + p)Vn+ – (V + V )Vn ) if n is even, = n– (Qn – (V + p)Un+ – (V + V )Un ) if n is odd. =
Note that, by taking p = in Theorem ., we obtain n– n n (Kn – Ln – Ln+ ) if n is even, (Qi ) = n– i (Qn – Fn – Fn+ ) if n is odd i=
Polatli and Kesim Advances in Difference Equations (2015) 2015:169
and n n n (Kn – Ln – Ln+ ) if n is even, (Ki ) = n+ i (Qn – Fn – Fn+ ) if n is odd, i= where Qi and Ki are the ith Fibonacci quaternion and the ith Lucas quaternion, respectively.
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors typed, read, and approved the final manuscript. Acknowledgements The authors would like to thank the referees for their valuable suggestions and comments. Received: 16 March 2015 Accepted: 18 May 2015 References 1. Vajda, S: Fibonacci and Lucas Numbers and the Golden Section. Ellis Horwood, Chichester (1989) 2. Carlitz, L: Some classes of Fibonacci sums. Fibonacci Q. 16, 411-426 (1978) 3. Carlitz, L, Ferns, HH: Some Fibonacci and Lucas identities. Fibonacci Q. 8, 61-73 (1970) 4. Hoggatt, VE: Some special Fibonacci and Lucas generating functions. Fibonacci Q. 9, 121-133 (1971) 5. Layman, JW: Certain general binomial-Fibonacci sums. Fibonacci Q. 15, 362-366 (1977) 6. Andrews, GE, Askey, R, Roy, R: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999) 7. Araci, S, Agyuz, E, Acikgoz, M: On a q-analog of some numbers and polynomials. J. Inequal. Appl. 2015, 19 (2015) 8. Graham, RL, Knuth, DE, Patashnik, O: Concrete Mathematics. Addison-Wesley, Reading (1994) 9. Kilic, E: Sums of the squares of terms of sequence {un }. Proc. Indian Acad. Sci. Math. Sci. 118, 27-41 (2008) 10. Kim, T, Dolgy, DV, Kim, DS, Rim, S-H: A note on the identities of special polynomials. Ars Comb. 113A, 97-106 (2014) 11. Simsek, Y: Special functions related to Dedekind-type DC-sums and their applications. Russ. J. Math. Phys. 17, 495-508 (2010) 12. Srivastava, HM, Choi, J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012) 13. Horadam, AF: Complex Fibonacci numbers and Fibonacci quaternions. Am. Math. Mon. 70, 289-291 (1963) 14. Ramirez, J: Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. An. S¸ tiin¸t. Univ. ‘Ovidius’ Constan¸ta, Ser. Mat. 23(2), 201-212 (2015) 15. Iakin, AL: Extended Binet forms for generalized quaternions of higher order. Fibonacci Q. 19, 410-413 (1981) 16. Iyer, MR: A note on Fibonacci quaternions. Fibonacci Q. 3, 225-229 (1969) 17. Swamy, MNS: On generalized Fibonacci quaternions. Fibonacci Q. 5, 547-550 (1973) 18. Iakin, AL: Generalized quaternions of higher order. Fibonacci Q. 15, 343-346 (1977) 19. Iakin, AL: Generalized quaternions with quaternion components. Fibonacci Q. 15, 350-352 (1977) 20. Horadam, AF: Quaternion recurrence relations. Ulam Q. 2, 23-33 (1993) 21. Halici, S: On Fibonacci quaternions. Adv. Appl. Clifford Algebras 22, 321-327 (2012) 22. Flaut, C, Shpakivskyi, V: On generalized Fibonacci quaternions and Fibonacci-Narayana quaternions. Adv. Appl. Clifford Algebras 23, 673-688 (2013)
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