Vol. 20 (2019), No. 2, pp. 807–821 DOI: 10.18514/MMN.2019.2935
ON THE GENERALIZED BI-PERIODIC FIBONACCI AND LUCAS QUATERNIONS
YOUNSEOK CHOO Received 11 April, 2019
Abstract. In this paper we introduce the generalized bi-periodic Fibonacci and Lucas quaternions which are the further generalizations of the bi-periodic Fibonacci and Lucas quaternions con- sidered in the literature. For those quaternions, we derive the generating functions, Binet’s for- mulas and Catalan’s identities.
2010Mathematics Subject Classification: 11B39; 11B37; 11B52
Keywords: generalized bi-periodic Fibonacci sequence, generalized bi-periodic Lucas sequence, quaternion, Binet’s formula, generating function, Catalan’s identity
1. INTRODUCTION
As is well known, the Fibonacci sequencefFngis generated from the recurrence relationFnDFn 1CFn 2 .n2/withF0D0; F1D1;and the Lucas sequence fLng is generated from the recurrence relation LnDLn 1CLn 2 .n2/ with L0D2; L1D1:The Binet’s formulas forfFngandfLngare respectively given by
FnD˛n ˇn
˛ ˇ ; LnD˛nCˇn;
where˛.> 0/andˇ.< 0/are roots of the equationx2 x 1D0.
Many authors generalized the Fibonacci and Lucas sequences by changing initial conditions and/or recurrence relations. In particular, Edson and Yayenie [5] intro- duced the bi-periodic Fibonacci sequencefpngdefined by
p0D0; p1D1; pnD
apn 1Cpn 2; ifnis even
bpn 1Cpn 2; ifnis odd .n2/: (1.1) The Binet’s formula forfpngis given by [5]
pnDa .nC1/
.ab/bn2c
˛n ˇn
˛ ˇ
; (1.2)
c 2019 Miskolc University Press
where˛.> 0/andˇ.< 0/are roots of the equationx2 abx abD0, and./is the parity function such that.n/D0ifnis even and.n/D1ifnis odd.
The bi-periodic Fibonacci sequencefpnggiven in (1.1) includes many sequences as special cases. ForaDbD1,fpngbecomes the Fibonacci sequence. ForaDbD 2,fpngbecomes the Pell sequence. IfaDbDk, thenfpngdenotes thek-Fibonacci sequence defined in [8], etc.
On the other hand, Bilgici [2] generalized the Lucas sequence by introducing the bi-periodic Lucas sequencefungdefined by
u0D2; u1Da; unD
bun 1Cun 2; ifnis even
aun 1Cun 2; ifnis odd .n2/: (1.3) IfaDb D1, thenfungbecomes the Lucas sequencefLng. IfaDb Dk, then fungbecomes thek-Lucas sequence in [7].
The Binet’s formula forfungis given by [2]
unD a .n/
.ab/bnC21c
.˛nCˇn/; (1.4)
where˛andˇare as defined in (1.2).
A quaternionqis defined by
qDq0e0Cq1e1Cq2e2Cq3e3;
whereq0; q1; q2; q32R,e0D1, ande1; e2ande3are the standard basis inR3such thatei2D 1,iD1; 2; 3;and
e1e2D e2e1De3; e2e3D e3e2De1; e3e1D e1e3De2:
As noted in the literature [1,6,9,20], quaternions are widely used in the fields of engineering and physics as well as mathematics, and attracted sustained attention from many researchers. In particular, a variety of results are available in the literature on the properties of quaternions related to the sequences described earlier. Horadam [13] defined the Fibonacci quaternion sequencefGngand Lucas quaternion sequence fHngas
GnDFne0CFnC1e1CFnC2e2CFnC3e3; HnDLne0CLnC1e1CLnC2e2CLnC3e3; whereFnandLnare respectively thenth Fibonacci and Lucas numbers.
Following the work of Horadam [13], diverse results have appeared in the literat- ure. Halici [10] obtained the generating functions, Binet’s formulas and some com- binatorial properties of the Fibonacci and Lucas quaternions. Halici [11] also intro- duced the complex Fibonacci quaternions. Ramirez [15] studied the properties of the k-Fibonacci andk-Lucas quaternions. C¸ imen and ˙Ipek [4], Szynal-Liana and Włoch [17] investigated the Pell and Pell-Lucas quaternions. Szynal-Liana and Włoch [17]
introduced the Jacobsthal and Jacobsthal-Lucas quaternions also. Catarino [3] con- sidered the modified Pell and modifiedk-Pell quaternions. Halici and Karatas¸ [12]
defined a general quaternion which includes several quaternions mentioned above as special cases.
Recently Tan et al. [19] introduced the bi-periodic Fibonacci quaternion sequence fPngdefined by
PnDpne0CpnC1e1CpnC2e2CpnC3e3; (1.5) wherepnis thenth bi-periodic Fibonacci number.
The Binet’s formula forfPngis given by [19]
PnD 8
<
:
1 .ab/bn2c
˛˛n ˇˇn
˛ ˇ
; ifnis even
1 .ab/bn2c
˛˛n ˇˇn
˛ ˇ
; ifnis odd (1.6)
where˛andˇare as defined in (1.2), and
˛D
3
X
lD0
a .lC1/
.ab/b2lc
˛lel;
ˇD
3
X
lD0
a .lC1/
.ab/b2lc ˇlel;
˛D
3
X
lD0
a .l/
.ab/blC21c
˛lel;
ˇD
3
X
lD0
a .l/
.ab/blC21c ˇlel:
Tan et al. [18] also introduced the bi-periodic Lucas quaternion sequencefUngas follows:
UnDune0CunC1e1CunC2e2CunC3e3; (1.7) whereunis thenth bi-periodic Lucas number.
The Binet’s formula forfUngis given by [18]
UnD 8
<
:
1 .ab/bnC21c
.˛˛nCˇˇn/; ifnis even
1 .ab/bnC21c
.˛˛nCˇˇn/; ifnis odd (1.8) where˛,ˇare as defined in (1.2), and˛,ˇ,˛andˇare as defined in (1.6).
If we use the initial conditionP0De1Ce2C2e3andP1De0Ce1C2e2C3e3
in (1.5), thenfPngis the same as the generalized Fibonacci quaternion sequence con- sidered in [14]. Also if we setP0D2e0Ce1C3e2C4e3andP1De0C3e1C4e2C
7e3in (1.5), thenfPngis the same as the generalized Lucas quaternion sequence con- sidered in [14].
In this paper we introduce the generalized bi-periodic Fibonacci and Lucas qua- ternion sequences which include fPngand fUng as special cases. For those qua- ternions, we derive the generating functions, Binet’s formulas and Catalan’s identit- ies.
2. MAIN RESULTS
2.1. Generalized bi-periodic Fibonacci quaternion
Consider the generalized bi-perioodic Fibonacci sequencefqngdefined by Sahin [16] and Yayenie [21] as
q0D0; q1D1; qnD
aqn 1Ccqn 2; ifnis even
bqn 1Cdqn 2; ifnis odd .n2/: (2.1) The Binet’s formula forfqngis given by [21]
qnDa .nC1/
.ab/bn2c
˛bn2c.˛Cd c/n bn2c ˇbn2c.ˇCd c/n bn2c
˛ ˇ
!
; (2.2)
where˛.> 0/andˇ.< 0/are roots of the equationx2 .abCc d /x abd D0:
Definition 1. We define the generalized bi-periodic Fibonacci quaternion sequence fQngby
QnDqne0CqnC1e1CqnC2e2CqnC3e3; (2.3) whereqnis thenth generalized bi-periodic Fibonacci number.
IfcDd D1, thenfQngbecomes the bi-periodic Fibonacci quaternion sequence given in (1.5).
IfaDbD1 andcDd D2, thenfQngbecomes the Jacobsthal quaternion se- quence defined in [17].
In the rest of the paper, we will use the following identities [21] whenever neces- sary: (i)˛CˇDabCc d;(ii)˛ˇD abd;(iii)˛.˛Cd c/Dab.˛Cd /;(iv) ˇ.ˇCd c/Dab.ˇCd /;(v).˛Cd /.ˇCd /Dcd:
Theorem 1 (Generating function). The generating function for the generalized bi-periodic Fibonacci quaternion sequence is
G.x/D 1 .abCd /x2Cbcx3
Q0Cx.1Cax cx2/Q1
1 .abCcCd /x2Ccdx4 : (2.4) Proof. We can show thatfQngsatisfies the same recurrence relation asfqngwith the initial condition
Q0Dq0e0Cq1e1Cq2e2Cq3e3
De1Cae2C.abCd /e3;
Q1Dq1e0Cq2e1Cq3e2Cq4e3
De0Cae1C.abCd /e2Ca.abCcCd /e3; and
Q2nD.abCcCd /Q2n 2 cdQ2n 4; Q2nC1D.abCcCd /Q2n 1 cdQ2n 3:
Then, proceeding as in the proof of [21, Theorem 7], we can obtain (2.4).
IfaDbandcDd, then
G.x/D .1 ax/Q0CxQ1
1 ax cx2
D xe0Ce1C.aCx/e2C.a2CcCacx/e3
1 ax cx2 :
Hence, for aDb Dc Dd D1, we get the generating function for the Fibonacci quaternion
G.x/Dxe0Ce1C.1Cx/e2C.2Cx/e3
1 x x2 ;
as in [10], and, foraDbDkandcDd D1, we obtain the generating function for thek-Fibonacci quaternion
G.x/Dxe0Ce1C.kCx/e2C.k2C1Ckx/e3
1 kx x2 ;
which is given in [15].
Theorem 2(Binet’s formula). The Binet’s formula for the generalized bi-periodic Fibonacci quaternion sequence is
QnD 8 ˆ<
ˆ:
1 .ab/bn2c
˛
e˛bn2c.˛Cd c/n bn2c ˇeˇbn2c.ˇCd c/n bn2c
˛ ˇ
; ifnis even
1 .ab/bn2c
˛o˛bn2c.˛Cd c/n bn2c ˇoˇbn2c.ˇCd c/n bn2c
˛ ˇ
; ifnis odd (2.5) where
˛eD
3
X
lD0
a .lC1/
.ab/bl2c
˛b2lc.˛Cd c/blC21cel;
ˇeD
3
X
lD0
a .lC1/
.ab/bl2c
ˇbl2c.ˇCd c/blC21cel;
˛oD
3
X
lD0
a .l/
.ab/blC21c
˛blC21c.˛Cd c/b2lcel;
ˇoD
3
X
lD0
a .l/
.ab/blC12 c
ˇblC21c.ˇCd c/b2lcel:
Proof. Firstly we note thatbn2c Dn .n/2 :
¿From (2.2) and (2.3), we have QnD a .nC1/
.ab/bn2c
˛bn2c.˛Cd c/n bn2c ˇbn2c.ˇCd c/n bn2c
˛ ˇ
! e0
C a .n/
.ab/ .n/.ab/bn2c
˛ .n/.˛Cd c/ .nC1/˛bn2c.˛Cd c/n bn2c
˛ ˇ
ˇ .n/.ˇCd c/ .nC1/ˇbn2c.ˇCd c/n bn2c
˛ ˇ
! e1
C a .nC1/
ab.ab/bn2c
˛.˛Cd c/˛bn2c.˛Cd c/n bn2c
˛ ˇ
ˇ.ˇCd c/ˇbn2c.ˇCd c/n bn2c
˛ ˇ
! e2
C a .n/
.ab/1C .n/.ab/bn2c
˛1C .n/.˛Cd c/1C .nC1/˛bn2c.˛Cd c/n bn2c
˛ ˇ
ˇ1C .n/.ˇCd c/1C .nC1/ˇbn2c.ˇCd c/n bn2c
˛ ˇ
! e3; or
QnD 1 .ab/bn2c
˛n˛bn2c.˛Cd c/n bn2c ˇnˇbn2c.ˇCd c/n bn2c
˛ ˇ
!
;
where
˛nDa .nC1/e0Ca .n/˛ .n/.˛Cd c/ .nC1/
.ab/ .n/ e1
Ca .nC1/˛.˛Cd c/
ab e2Ca .n/˛1C .n/.˛Cd c/1C .nC1/
.ab/1C .n/ e3; ˇnDa .nC1/e0Ca .n/ˇ .n/.ˇCd c/ .nC1/
.ab/ .n/ e1
Ca .nC1/ˇ.ˇCd c/
ab e2Ca .n/ˇ1C .n/.ˇCd c/1C .nC1/
.ab/1C .n/ e3:
Ifnis even, then
˛nDae0C.˛Cd c/e1Ca˛.˛Cd c/
ab e2C˛.˛Cd c/2 ab e3
D
3
X
lD0
a .lC1/
.ab/b2lc
˛b2lc.˛Cd c/blC21cel;
ˇnDae0C.ˇCd c/e1Caˇ.ˇCd c/
ab e2Cˇ.ˇCd c/2 ab e3
D
3
X
lD0
a .lC1/
.ab/b2lc
ˇb2lc.ˇCd c/blC21cel:
Similarly, ifnis odd, then
˛nD
3
X
lD0
a .l/
.ab/blC21c
˛blC21c.˛Cd c/b2lcel;
ˇnD
3
X
lD0
a .l/
.ab/blC21c
ˇblC21c.ˇCd c/bl2cel;
and the proof is completed.
IfcDd D1, then (2.5) becomes the Binet’s formula for the bi-periodic Fibonacci quaternion given in (1.6).
Theorem 3 (Catalan’s identity). The Catalan’s identity for the generalized bi- periodic Fibonacci quaternion sequence is
Q2n QnC2rQn 2r
D 8 ˆˆ ˆˆ
<
ˆˆ ˆˆ :
.cd /n 2r2 ˛eˇe .˛Cd /
2r .cd /r
Cˇe˛e .ˇCd /2r .cd /r
.˛ ˇ /2
!
; if n i s eve n;
abc.cd /n 2r 12 ˛oˇo .cd /
r .˛Cd /2r
Cˇo˛o .cd /r .ˇCd /2r
.˛ ˇ /2
!
; if n i s od d:
(2.6) Proof. Firstly, assume thatnis even, and let
X1D.˛ ˇ/2.ab/nQ2n;
X2D.˛ ˇ/2.ab/nQnC2rQn 2r: Then
X1D
˛e˛n2.˛Cd c/n2 ˇeˇn2.ˇCd c/n22
D˛2e˛n.˛Cd c/nCˇe2ˇn.ˇCd c/n
.˛eˇeCˇe˛e/˛n2.˛Cd c/n2ˇn2.ˇCd c/n2; D˛2e˛n.˛Cd c/nCˇe2ˇn.ˇCd c/n
.˛eˇeCˇe˛e/.ab/n.˛Cd /n2.ˇCd /n2; D˛2e˛n.˛Cd c/nCˇe2ˇn.ˇCd c/n
.˛eˇeCˇe˛e/.ab/n.cd /n2; and
X2D
˛e˛nC22r.˛Cd c/nC22r ˇeˇnC22r.ˇCd c/nC22r
˛e˛n22r.˛Cd c/n22r ˇeˇn22r.ˇCd c/n22r D˛2e˛n.˛Cd c/nCˇe2ˇn.ˇCd c/n
˛eˇe˛nC22r.˛Cd c/nC22rˇn22r.ˇCd c/n22r ˇe˛e˛n22r.˛Cd c/n22rˇnC22r.ˇCd c/nC22r; D˛2e˛n.˛Cd c/nCˇe2ˇn.ˇCd c/n
˛eˇe.ab/n.˛Cd /nC2r2 .ˇCd /n22r ˇe˛e.ab/n.˛Cd /n22r.ˇCd /nC22r; D˛2e˛n.˛Cd c/nCˇe2ˇn.ˇCd c/n
˛eˇe.ab/n.cd /n22r.˛Cd /2r ˇe˛e.ab/n.cd /n22r.ˇCd /2r: Hence
X1 X2D˛eˇe.ab/n.cd /n22r
.˛Cd /2r .cd /r Cˇe˛e.ab/n.cd /n22r
.ˇCd /2r .cd /r
; and the proof is completed for the case wherenis even.
Whennis odd, we can proceed similarly, and details are omitted.
IfcDd D1, then˛2Dab.˛C1/and
.˛C1/2r 1D.ab/2r.˛C1/2r .ab/2r .ab/2r
D˛4r .ab/2r .ab/2r :
Similarly
.ˇC1/2r 1D ˇ4r .ab/2r .ab/2r ; and Theorem3reduces to [19, Theorem 5].
2.2. Generalized bi-periodic Lucas quaternion
Consider the generalized bi-periodic Lucas sequencefvngdefined by Bilgici [2]
as
v0DdC1
d ; v1Da; vnD
bvn 1Cdvn 2; ifnis even
avn 1Ccvn 2; ifnis odd .n2/: (2.7) The Binet’s formula forfvngis given by [2]
vnD a .n/
.ab/bn 12 c
.˛CdC1/˛bn 12 c.˛Cd c/bn2c .ˇCdC1/ˇbn 12 c.ˇCd c/bn2c
˛ ˇ
!
; (2.8) where˛andˇare as defined in (2.2).
Definition 2. The generalized bi-periodic Lucas quaternion sequencefVngis defined by
VnDvne0CvnC1e1CvnC2e2CvnC3e3; (2.9) wherevnis thenth generalized bi-periodic Lucas number.
IfcDd D1, fVngbecomes the bi-periodic Lucas quaternion sequence given in (1.7).
Theorem 4 (Generating function). The generating function for the generalized bi-periodic Lucas quaternion sequence is
H.x/D 1 .abCc/x2Cadx3
V0Cx.1Cbx dx2/V1
1 .abCcCd /x2Ccdx4 : (2.10)
Proof. Replacing Q0, Q1, a, b, c andd by V0, V1, b, a, d andc in (2.4), we
obtain (2.10).
IfaDbandcDd, then
H.x/D .1 ax/V0CxV1
1 ax cx2 D
cC1 ax
c e0C aC.cC1/x
e1C.a2CcC1Cacx/e2
1 ax cx2
C a3C2acCaC.a2Cc2C1/x e3
1 ax cx2 :
Hence, for aDb Dk and c Dd D1, we obtain the generating function for the k-Lucas quaternion
H.x/D.2 kx/e0C.kC2x/e1C.k2C2Ckx/e2C k3C3kC.k2C2/x e3
1 kx x2 ;
as in [15].
Theorem 5(Binet’s formula). The Binet’s formula for the generalized bi-periodic Lucas quaternion sequence is
VnD 8
<
:
1 .ab/bn21c
Vne; ifnis even
1 .ab/bn21c
Vno; ifnis odd (2.11)
where
Vne D˛o.˛CdC1/˛bn 12 c.˛Cd c/bn2c ˇo.ˇCdC1/ˇbn 12 c.ˇCd c/bn2c
˛ ˇ ;
VnoD˛e.˛CdC1/˛bn 12 c.˛Cd c/bn2c ˇe.ˇCdC1/ˇbn 12 c.ˇCd c/bn2c
˛ ˇ ;
with˛e,ˇe,˛oandˇoas defined in (2.5).
Proof. Using the Binet’s formula forfvngand proceeding as in the proof of The-
orem2, we can easily obtain (2.11).
IfcDd D1, then
.˛C2/˛bn21cCbn2cD.˛C2/˛n 1 D
1C2
˛
˛n
D 1 2ˇ
ab
˛n
D.˛ ˇ/˛n ab ; and
.ˇC2/ˇbn21cCbn2cD.ˇC2/ˇn 1 D
1C2 ˇ
ˇn
D 1 2˛
ab
ˇn
D.ˇ ˛/ˇn ab :
Hence (2.11) reduces to the Binet’s formula for the bi-periodic Lucas quaternion given in (1.8).
We verify (2.11) fornD1. From (2.7) and the definition offVng, we have V1Dv1e0Cv2e1Cv3e2Cv4e3
Dae0C.abCdC1/e1Ca.abCcCdC1/e2
C.a2b2CabcC2abdCabCd2Cd /e3: On the other hand, ifnD1, then (2.11) becomes
V1D˛e.˛CdC1/ ˇe.ˇCdC1/
˛ ˇ :
In this case,˛e andˇe respectively can be written as
˛eDaeoC.˛Cd c/e1Ca.˛Cd /e2C.˛Cd /.˛Cd c/e3; ˇeDaeoC.ˇCd c/e1Ca.ˇCd /e2C.ˇCd /.ˇCd c/e3: Let
˛e.˛CdC1/ ˇe.ˇCdC1/DE0e0CE1e1CE2e2CE3e3: Then
E0Da.˛ ˇ/;
E1D.˛CdC1/.˛Cd c/ .ˇCdC1/.ˇCd c/
D.˛Cd /2 .ˇCd /2 .c 1/.˛ ˇ/
D.˛CˇC2d /.˛ ˇ/ .c 1/.˛ ˇ/
D.abCdC1/.˛ ˇ/;
E2Da .˛Cd /.˛Cd c/ .ˇCd /.ˇCd c/
Da .˛Cd /2 .ˇCd /2C.˛ ˇ/
Da .˛CˇC2d /.˛ ˇ/C.˛ ˇ/
Da.abCcCdC1/.˛ ˇ/;
E3D.˛Cd /.˛CdC1/.˛Cd c/ .ˇCd /.ˇCdC1/.ˇCd c/
D.˛Cd /3 .ˇCd /3 .c 1/ .˛Cd /2 .ˇCd /2
c.˛ ˇ/
D .˛CˇC2d /2 .˛Cd /.ˇCd / .˛ ˇ/
.c 1/.˛CˇC2d /.˛ ˇ/ c.˛ ˇ/
D .abCcCd /2 cd
.˛ ˇ/ .c 1/.abCcCd /.˛ ˇ/ c.˛ ˇ/
D.a2b2CabcC2abdCabCd2Cd /.˛ ˇ/:
Hence (2.11) is true fornD1.
Theorem 6 (Catalan’s identity). The Catalan’s identity for the generalized bi- periodic Lucas quaternion sequence is
Vn2 VnC2rVn 2r
D 8 ˆˆ ˆˆ ˆˆ ˆ<
ˆˆ ˆˆ ˆˆ ˆ:
.cd /n22r.d˛ ˇ cdCd2/.dˇ ˛ cdCd2/ d2
˛oˇo .˛Cd /2r .cd /r
Cˇo˛o .ˇCd /2r .cd /r
.˛ ˇ /2
; if n i s eve n;
.cd /n 2r2 1.d˛ ˇ cdCd2/.dˇ ˛ cdCd2/ abd
˛eˇe .cd /r .˛Cd /2r
Cˇe˛e .cd /r .ˇCd /2r
.˛ ˇ /2
; if n i s od d:
(2.12)
Proof. Assume thatnis even, and let
Y1D.˛ ˇ/2.ab/n 2Vn2;
Y2D.˛ ˇ/2.ab/n 2VnC2rVn 2r: Then
Y1D
˛o.˛CdC1/˛n22.˛Cd c/n2 ˇo.ˇCdC1/ˇn22.ˇCd c/n22
D˛2o.˛CdC1/2˛n 2.˛Cd c/nCˇ2o.ˇCdC1/2ˇn 2.ˇCd c/n .˛oˇoCˇo˛o/.˛CdC1/.ˇCdC1/˛n22.˛Cd c/n2ˇn22.ˇCd c/n2; and
Y2D
˛o.˛CdC1/˛nC2r2 2.˛Cd c/nC22r ˇo.ˇCdC1/ˇnC2r2 2.ˇCd c/nC22r
˛o.˛CdC1/˛n 2r2 2.˛Cd c/n22r ˇo.ˇCdC1/ˇn 2r2 2.ˇCd c/n22r
; D˛o2.˛CdC1/2˛n 2.˛Cdc/nCˇo2.ˇCdC1/2ˇn 2.ˇCd c/n
˛oˇo.˛CdC1/.ˇCdC1/˛nC2r2 2.˛Cd c/nC22rˇn 2r2 2.ˇCd c/n22r; ˇo˛o.˛CdC1/.ˇCdC1/˛n 2r2 2.˛Cd c/n22rˇnC2r2 2.ˇCd c/nC22r: Hence
Y1 Y2D˛oˇoA1Cˇo˛oA2; where
A1D.˛CdC1/.ˇCdC1/
˛nC2r2 2.˛Cd c/nC22rˇn 2r2 2.ˇCd c/n22r
˛n22.˛Cd c/n2ˇn22.ˇCd c/n2 A2D.˛CdC1/.ˇCdC1/
˛n 2r2 2.˛Cd c/n22rˇnC2r2 2.ˇCd c/nC22r
˛n22.˛Cd c/n2ˇn22.ˇCd c/n2 :
Since˛ˇD abd and˛CˇDabCc d, we have .˛CdC1/D
1CdC1
˛
˛
D
1 .dC1/ˇ abd
˛
D
.˛Cˇ/d d.c d / .dC1/ˇ abd
˛
D.d˛ ˇ cdCd2/˛
abd :
Similarly
.ˇCdC1/D.dˇ ˛ cdCd2/ˇ
abd :
Then
A1D.d˛ ˇ cdCd2/.dˇ ˛ cdCd2/˛n2.˛Cd c/n2ˇn2.ˇCd c/n2 .abd /2
˛r.˛Cd c/r ˇr.ˇCd c/r 1
D.ab/n 2.cd /n2.d˛ ˇ cdCd2/.dˇ ˛ cdCd2/ d2
˛2r.˛Cd c/2r ˛r.˛Cd c/rˇr.ˇCd c/r
˛r.˛Cd c/rˇr.ˇCd c/r
D.ab/n 2.cd /n22r.d˛ ˇ cdCd2/.dˇ ˛ cdCd2/ .˛Cd /2r .cd /r
d2 :
Similarly we have
A2D.ab/n 2.cd /n22r.d˛ ˇ cdCd2/.dˇ ˛ cdCd2/ .ˇCd /2r .cd /r
d2 ;
and the proof is completed for the case wherenis even.
Using the same procedure, we can also prove (2.12) for the case wherenis odd.
IfcDd D1, then Theorem6reduces to [18, Theorem 5].
3. CONCLUSIONS
In this paper we introduced the generalized bi-periodic Fibonacci and Lucas qua- ternions which are the further generalizations of the bi-periodic Fibonacci and Lucas quaternions considered in the literature. For those quaternions, we obtained the gen- erating functions, Binet’s formulas and Catalan’s identities.
ACKNOWLEDGEMENT
The author thanks to the anonymous reviewer for helpful comments which led to improved presentation of the paper.
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Author’s address
Younseok Choo
Hongik University, Department of Electronic and Electrical Engineering, 2639 Sejong-Ro, 30016 Sejong, Republic of Korea
E-mail address:yschoo@hongik.ac.kr
