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Journal of Integer Sequences, Vol. 18 (2015),
Article 15.8.2
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47
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23 11
A Note on Catalan’s Identity for the
k-Fibonacci Quaternions
Emrah Polatlı and Seyhun Kesim
Department of Mathematics
Bülent Ecevit University
67100 Zonguldak
Turkey
emrahpolatli@gmail.com
seyhun.kesim@beun.edu.tr
Abstract
Ramı́rez recently conjectured a version of Catalan’s identity for the k-Fibonacci
quaternions. In this note we give a proof of (a suitably reformulated version of) this
identity.
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Introduction
For any positive real number k, define the k-Fibonacci and k-Lucas sequences, (Fk,n )n∈N and
(Lk,n )n∈N , as follows:
Fk,0 = 0, Fk,1 = 1, and Fk,n = kFk,n−1 + Fk,n−2 , n ≥ 2,
and
Lk,0 = 2, Lk,1 = k, and Lk,n = kLk,n−1 + Lk,n−2 , n ≥ 2,
respectively.
Let α and β be the roots of the characteristic equation x2 − kx − 1 = 0. Then the Binet
formulas for the k-Fibonacci and k-Lucas sequences are
Fk,n =
αn − β n
α−β
1
and
Lk,n = αn + β n ,
√
√
where α = k + k 2 + 4 /2 and β = k − k 2 + 4 /2.
A quaternion p, with real components a0 , a1 , a2 , a3 and basis 1, i, j, k, is an element of
the form
p = a0 + a1 i + a2 j + a3 k = (a0 , a1 , a2 , a3 ) , (a0 1 = a0 ) ,
where
i2 = j 2 = k2 = −1,
ij = k = −ji, jk = i = −kj, ki = j = −ik.
Ramı́rez [1] defined the nth k-Fibonacci quaternion, Dk,n , as follows:
Dk,n = Fk,n + Fk,n+1 i + Fk,n+2 j + Fk,n+3 k, n ≥ 0,
where Fk,n is the nth k-Fibonacci number. Ramı́rez [1] also gave the Binet formula for the
k-Fibonacci quaternion as follows:
Dk,n =
b n
α
bαn − ββ
,
α−β
where α
b = 1 + αi + α2 j + α3 k and βb = 1 + βi + β 2 j + β 3 k.
Ramı́rez [1] conjectured that the Catalan identity for the k-Fibonacci quaternions is
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Dk,n−r Dk,n+r − Dk,n
= (−1)n−r (2Fk,r Dk,r − Gk,r k) ,
for n ≥ r ≥ 1, where Gk,r is the sequence satisfying the following recurrence:
Gk,0 = 0, Gk,1 = k 2 + 2k, and Gk,n = k 2 + 2 Gk,n−1 − Gk,n−2 , n ≥ 2.
However, in this short paper, we show that this conjecture is incorrect, by giving the correct
Catalan identity and proving it.
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Catalan identity for the k-Fibonacci quaternions
We need the following lemma.
Lemma 1. For r ≥ 1, we have
b r − βbα
α
bββ
b αr
= −2Dk,r + Lk,2 Lk,r k.
α−β
2
Proof. Since
α
bβb = 2 + 2βi + 2β 2 j + α3 + β 3 + α − β k
and
βbα
b = 2 + 2αi + 2α2 j + α3 + β 3 + β − α k,
we get
b r − βbα
α
bββ
b αr
= −2Fk,r − 2Fk,r+1 i − 2Fk,r+2 j + (−2Fk,r+3 + Lk,2 Lk,r ) k
α−β
= −2Dk,r + Lk,2 Lk,r k.
Theorem 2. For n ≥ r ≥ 1, Catalan identity for the k-Fibonacci quaternions is
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Dk,n−r Dk,n+r − Dk,n
= (−1)n−r+1 (2Fk,r Dk,r − Lk,2 Fk,2r k) .
Proof. By considering the Binet formula for the k-Fibonacci quaternions, quaternion multiplication and Lemma 1, we obtain
!
!
!2
n−r
n+r
n
b n−r
b n+r
b n
α
b
α
−
ββ
α
b
α
−
ββ
α
b
α
−
ββ
2
Dk,n−r Dk,n+r − Dk,n
=
−
α−β
α−β
α−β
r
(αβ)n
αr
bα
b 1− β
=
+
β
b
1
−
α
b
β
αr
βr
(α − β)2
!
b
αr − β r α
bβb βbα
− r
= (αβ)n
2
r
α
β
(α − β)
!
r
r
b r − βbα
α
bββ
b αr
n−r α − β
= (αβ)
α−β
α−β
= (αβ)n−r Fk,r (−2Dk,r + Lk,2 Lk,r k)
= (−1)n−r+1 (2Fk,r Dk,r − Lk,2 Fk,2r k) .
References
[1] J. L. Ramı́rez, Some combinatorial properties of the k -Fibonacci and the k-Lucas quaternions, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 23 (2015), 201–212.
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2010 Mathematics Subject Classification: Primary 11B39.
Keywords: k-Fibonacci quaternion, Catalan’s identity.
Received May 18 2015; revised version received May 20 2015; June 18 2015; June 24 2015.
Published in Journal of Integer Sequences, July 16 2015.
Return to Journal of Integer Sequences home page.
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