Horadam
2
k-ions
Melih Göcen and Yüksel Soykan
Abstract. In this paper, we generalize Fibonacci quaternion, octonion, sedenion, trigintaduonion, etc. and de…ne Horadam2k-ions and investigate their properties. Each Horadam (such as Fibonacci, Lucas, Pell) quaternions, octonions and sedenions are Horadam 2k-ions. We also present connection to some earlier works.
Key Words: quaternions, octonions, sedenions, Horadam numbers, Fibonacci quaternion, Cayley-Dickson algebra.
2010 Mathematics Subject Classi…cation: 11B39, 11B83.
1. Introduction
In this work we generalize Fibonacci quaternion, octonion, sedenion, trigintaduonion and so on. In section 1 we give de…nition of 2k-ions and in section 2 we give de…nition of Horadam numbers which is generalization of Fibonacci numbers and contains Fibonacci, Lucas, Pell, Jacobsthal and all other second order sequences. In section 3, we introduce Horadam2k-ions which contains Fibonacci, Lucas, Pell, Jacob-sthal quaternions (octonions and sedenions). Moreover, in that section we give Binet formula, generating function, norm value of Horadam2k-ions and also Catalan identity, Cassini identity, d’Ocagne identity and a summation formula. In section 4 we present matrix methods in Horadam2k-ions.
In the following, we will brie‡y present important and very known algebra: Cayley-Dickson algebra. In 1843, William Rowan Hamilton discovered the quaternions (H), which is a4 = 22-dimensional algebra
overR. This algebra is an associative and a noncommutative algebra. In 1843, John Graves discovered the octonions (O), a8 = 23-dimensional algebra overRwhich is a nonassociative and a noncommutative algebra.
In 1845, these algebras were rediscovered by Arthur Cayley. They are also known as the Cayley numbers. This process, of passing fromRtoC, fromCtoHand fromHtoOwas generalized to algebras over arbitrary …elds and rings. It is called the Cayley-Dickson doubling process or the Cayley–Dickson process. The next doubling process applied toOthen produces an algebraS(dim24= 16) called the sedenions. This doubling
process can be extended beyond the sedenions to form what are known as the2k-ions (see for example [1], [18], [22]).
Now, we explain this doubling process. The Cayley-Dickson algebras are a sequenceA0; A1; :::of
non-associativeR-algebras with involution. The term “conjugation”can be used to refer to the involution because it generalizes the usual conjugation on the complex numbers. For a full explanation of the basic properties of Cayley-Dickson algebras, see [1]. Cayley-Dickson algebras are de…ned inductively. We begin by de…ningA0
to beR:Given Ak 1;the algebraAk is de…ned additively to beAk 1 Ak 1:Conjugation inAk is de…ned by
(a; b) = (a; b)
and multiplication is de…ned by
(1.1) (a; b)(c; d) = (ac db; da+bc)
and addition is de…ned by componentwise as
(a; b) + (c; d) = (a+c; b+d):
Ak has dimension N = 2k as an R vector space. If we set, as usual, kxk =
p
Re(xx) for x 2 Ak then xx=xx=kxk2:
For speci…ck;how2k-ions are called, is given in the following table. Table 1.
k 2 3 4 5 :::
2k-ions Quaternions Octonions Sedenions Trigintaduonions :::
Now, for a …xedksuppose thatBN =fei2Ak:i= 0;1;2; :::; N 1gis the basis forAk, whereN= 2k is the dimension ofAk; e0is the identity (or unit) ande1; e2; :::; eN 1 are called imaginaries. Then a2k-ions
S2Ak can be written as
S =
NX1
i=0
aiei=a0+
NX1
i=1
aiei
where a0; a1; :::; aN 1 are all real numbers. Herea0 is called the real part ofS and
PN 1
i=1 aiei is called its imaginary part.
Addition of2k-ions is de…ned as componentwise and multiplication is de…ned as follows: ifS
1; S22Ak
then we have
(1.2) S1S2=
NX1
i=0
aiei
! N 1 X
i=0
biei
!
=
NX1
i;j=0
aibj(eiej):
The operations requiring for the multiplication in (1.2) are quite a lot. At the hypercomplex algebra the most time-consuming operation is the multiplication of two hypercomplex numbers. This is because the multiplication of twoN-dimensional hypercomplex numbers requiresN2real multiplications andN(N 1)
can compute the same result in only 498real multiplications (or multipliers – in hardware implementation case) and943 real additions, for details see [5].
Using direct multiplication, the number of the operations requiring for the multiplication of two (speci…c)
2k-ions is given in the following table. Table 2.
2k-ions Computation method Multiplications Additions Quaternions Based on expression (1.2) 16 12
Octonions Based on expression (1.2) 64 56
Sedenions Based on expression (1.2) 256 240
Trigintaduonions Based on expression (1.2) 1024 992
E¢ cient algorithms for the multiplication of quaternions, octonions, sedenions, and trigintaduonions with reduced number of real multiplications is already exist and results of synthesizing an e¢ cient algorithm of computing the two2k-ions product are presented in the following table.
Table 3.
2k-ions Computation method Multiplications Additions
Quaternions Algorithm in [24] 8
Octonions Algorithm in [3] 32 88
Sedenions Algorithm in [4] 122 298
Trigintaduonions Algorithm in [5] 498 943
2. Horadam Numbers
Now let us recall the de…nition of Horadam numbers. In 1965, Horadam [15] de…ned a generalization of Fibonacci sequence, that is, he de…ned a second-order linear recurrence sequencefWn(a; b;p; q)g, or simply fWng, as follows:
(2.1) Wn=pWn 1+qWn 2; W0=a; W1=b; (n 2)
where a; b; p and q are arbitrary real numbers, see also Horadam [13], [16] and [17], . Now these numbers fWn(a; b;p; q)g are called Horadam numbers.
For some speci…c values ofa; b; pandq, it is worth presenting these special Horadam numbers in a table as a speci…c name.
Name of sequence Wn(a; b;p; q) Binet Formula
Fibonacci Wn(0;1; 1;1) =Fn
1+p5 2
n
1 p5 2
n
p
5
Lucas Wn(2;1; 1;1) =Ln 1+
p 5 2
n
+ 1 2p5 n
Pell Wn(0;1; 2;1) =Pn
1 +p2 n 1 p2 n
2p2
Pell-Lucas Wn(2;2; 2;1) =pn 1 + p
2 n+ 1 p2 n
Jacobsthal Wn(0;1; 1;2) =Jn 2
n
( 1)n 3
Jacobsthal-Lucas Wn(2;1; 1;2) =jn 2n+ ( 1)n
We can list some important properties of Horadam numbers that are needed.
Binet formula of Horadam sequence can be calculated using its characteristic equation which is given as
t2 pt q= 0:
The roots of characteristic equation are
=p+
p
2 ; =
p p
2 :
where =p2+ 4q: Using these roots and the recurrence relation Binet formula can be given as
follows
(2.2) Wn=
A n B n
where A=b a and B=b a :
The generating function for Horadam numbers is
(2.3) g(t) =W0+ (W1 pW0)t
1 pt qt2 :
The Cassini identity for Horadam numbers is
(2.4) Wn+1Wn 1 Wn2=qn 1(pW0W1 W12 W02q):
A summation formula for Horadam numbers is
(2.5)
n
X
i=0
Wi=
W1 W0(p 1) +qWn Wn+1
1 p q :
For =p2+ 4q >0; and are reals and 6= : Note also that
(2.6) 2= p q
and
A n = Wn+qWn 1;
B n = Wn+qWn 1:
3. Horadam 2k-ions
In this section we introduce Horadam 2k-ions and present their important properties. We give Binet formula, generating function, Cassini identity, summation formula and norm of these2k-ions. First, we give some information about (Horadam) quaternion, octonion, sedenion and trigintaduonion sequences from the literature.
Horadam [14] introducednth Fibonacci andnth Lucas quaternions as
Qn=Fn+Fn+1e1+Fn+2e2+Fn+3e3= 3
X
s=0
Fn+ses
and
Rn =Ln+Ln+1e1+Ln+2e2+Ln+3e3= 3
X
s=0
Ln+ses
respectively, where Fn and Ln are the nth Fibonacci and Lucas numbers respectively. See also Halici and Karata¸s [12], and Polatl¬[23].
Various families of octonion number sequences (such as Fibonacci octonion, Pell octonion, Jacob-sthal octonion, and third order JacobJacob-sthal octonion) have been de…ned and studied by a number of authors in many di¤erent ways. For example, Keçilioglu and Akku¸s [21] introduced the Fibonacci and Lucas octonions as
Qn=
7
X
s=0
Fn+ses
and
Rn =
7
X
s=0
Ln+ses
respectively, where Fn and Ln are the nth Fibonacci and Lucas numbers respectively. In [10], Çimen and ·Ipek introduced Jacobsthal octonions and Jacobsthal-Lucas octonions. See also Tasci [26] for k-Jacobsthal and k-Jacobsthal-Lucas Quaternions and see ·Ipek and Çimen [19] and Karata¸s and Halici [20] for Horadam type octonions. Moreover, we refer to Szynal-Liana and Wlock [25] for Pell quaternions and Pell octonions and Catarino [6] for Modi…ed Pell and Modi…ed k-Pell Quaternions and Octonions.
sedenions as
b Fn=
15
X
s=0
Fn+ses
and
b Ln=
15
X
s=0
Ln+ses
respectively, where Fn and Ln are the nth Fibonacci and Lucas numbers respectively. In [7], Catarino introduced k-Pell and k-Pell–Lucas sedenions. In [9], Çimen and ·Ipek introduced Jacob-sthal and JacobJacob-sthal-Lucas sedenions.
Gül [11] introduced the k-Fibonacci and k-Lucas trigintaduonions as
T Fk;n=
31
X
s=0
Fk;n+ses
and
T Lk;n=
31
X
s=0
Lk;n+ses
respectively, whereFk;nandLk;nare thenth k-Fibonacci and k-Lucas numbers respectively.
Definition1. The Horadam2k-ions sequence
n c Wn
o
n 0 is de…ned by the following recurrence relation:
(3.1) cWn=
NX1
s=0
Wn+ses
whereWn is thenthgeneralized Horadam number.
Note that from the de…nition of (2.1) and (3.1) we have the following recurrence relation:
(3.2) cWn=pWcn 1+qcWn 2:
Firstly, we present the Binet formula. This formula is very useful for …nding desired Horadam2k-ions and takes part at many theorems’proof. From now on, for …xedk(so that the dimension of Horadam2k-ions is2k =N) we …xed the following notations:
b=
NX1
s=0
se s; b=
NX1
s=0
se s:
Theorem 1. The Binet formula for Horadam 2k-ions is
(3.3) Wcn =
Proof. Repeated use of (2.2) in (3.1) enable us to write:
c Wn =
NX1
s=0
Wn+ses=
A n B n e0+
A n+1 B n+1
e1
+ A
n+2 B n+2
e2+:::+
A n+N 1 B n+N 1! eN 1
= Ab
n Bb n :
Alternative Proof of Theorem 1: We use the following identities:
A n = Wn+qWn 1;
B n = Wn+qWn 1:
Note that
c
Wn+qWcn 1 = (Wn+Wn+1e1+:::+Wn+N 1eN 1) +q(Wn 1+Wne1+:::+Wn+N 2eN 1)
= ( Wn+qWn 1)e0+ ( Wn+1+qWn)e1+:::+ ( Wn+N 1+qWn+N 2)eN 1
From the identityA n= W
n+qWn 1 fornth Horadam numberWn;we have
(3.4) cWn+qcWn 1=Ab n:
Similarly, we obtain
(3.5) cWn+qWcn 1=Bb n
Substracting (3.5) from (3.4), we have
(3.6) Wcn cWn=Ab n Bb n:
So this proves (3.3).
For some particular cases of Binet formulas for Horadam2k-ions, see Tables 5,6,7 and 8. It is well known that for Horadam 2k-ions Wc
n de…ned by (3.1) the ordinary generating function is g(t) =P1n=0Wcntn:In the following theorem we present the generating function for Horadam2k-ions.
Theorem 2. The generating function for Horadam 2k-ions is
(3.7) g(t) =Wc0+ (Wc1 pWc0)t
1 pt qt2 :
Proof. Let
g(t) =
1
X
n=0
be generating function of Horadam 2k-ions. Then using de…nition of Horadam 2k-ions, and substracting ptg(t)andqt2g(t)fromg(t)we obtain (note the shift in the indexnin the third line)
(1 pt qt2)g(t) =
1
X
n=0
c
Wntn pt
1
X
n=0
c
Wntn qt2
1
X
n=0
c Wntn
=
1
X
n=0
c Wntn p
1
X
n=0
c
Wntn+1 q
1
X
n=0
c Wntn+2
=
1
X
n=0
c Wntn p
1
X
n=1
c
Wn 1tn q 1
X
n=2
c Wn 2tn
= (cW0+cW1t) pWc0t+ 1
X
n=2
c
Wn pcWn 1 qcWn 2 tn
= cW0+ (cW1 pWc0)t:
Rearranging above equation, we get
g(t) =Wc0+ (Wc1 pWc0)t
1 pt qt2 :
For Horadam 2k-ions Wc
n de…ned by (3.1), the exponential generating function is de…ned by e(t) =
P1
n=0cWnt
n
n!:
Theorem 3. ForWcn, we have
(3.8) e(t) = Abe
t Bbe t :
Proof. Using (3.3) ine(t) =P1n=0Wcnt
n
n!, we obtain
(3.9) e(t) =
1
X
n=0
Ab n Bb n!tn n!:
Combining the formulaet=P1 n=0t
n
n! and (3.9), we get (3.8).
There are three well-known identities for Fibonacci type numbers, namely, Catalan’s, Cassini’s, and d’Ocagne’s identities. The proofs of these identities are based on Binet formulas. We can obtain these types of identities for Horadam2k-ions using the Binet formulas derived above. Two di¤erent Catalan’s identities for Horadam2k-ions are given in the next theorem.
Theorem 4 (Catalan’s Identity). For any integersmandn we have
(a):
(3.10) Wcm ncWm+n Wcm2 =
AB( )m( n n)( nbb nbb)
( )2
(b):
(3.11) Wcm+ncWm n Wcm2 =
AB( )m( n n)( nbb nbb)
Proof. By using Binet formula, we obtain
c
Wm nWcm+n Wcm2 =
Ab m n Bb m nAb m+n Bb m+n Ab m Bb m!2 :
After necessary calculations we get the identity (3.10). Similarly we obtain (3.11).
Taking n = 1 in Catalan’s identitity we obtain Cassini’s identitity for Horadam 2k-ions. In the next theorem, we state two di¤erent Cassini identities.
Corollary1 (Cassini’s Identitity). For any integerm;we have
(a):
(3.12) Wcm 1Wcm+1 cWm2 =
AB( )m 1( bb bb)
;
(b):
(3.13) Wcm+1cWm 1 cWm2 =
AB( )m 1( bb bb)
:
Note that from non-commutativity bb6=bb and so bb bb6=bb( ).
We emphasize some particular cases of Binet formulas, generating functions and Cassini’s identities for Horadam2k-ions:
Leta= 0; b=p=q= 1:Fork= 2; k= 3; k= 4 andk= 5we get the Binet formula, generating function, Cassini’s identitity for Fibonacci quaternions which is given by Halici and Karata¸s in [12], the Binet formula, generating function, Cassini’s identitity for Fibonacci octonions which is given by Keçilio¼glu and Akku¸s in [21] or Halici and Karata¸s [12], the Binet formula, generating function, Cassini’s identitity for Fibonacci sedenions which is given by Bilgici, Toke¸ser and Ünal [2], and the Binet formula, generating function, Cassini’s identitity for Fibonacci trigintaduonions which is given by Gül in [11], respectively.
Leta= 0; b=p= 1; q= 2:Fork= 2; k= 3;andk= 4, we obtain the Binet formula, generating function, Cassini’s identitity for Jacobsthal quaternions which is given by Tasci in [26], the Binet formula, generating function, Cassini’s identitity for Jacobsthal octonions which is given by Çimen and ·Ipek in [10] and the Binet formula, generating function, Cassini’s identitity for Jacobsthal sedenions which is given by Çimen and ·Ipek in [9], respectively.
We give the next four tables to show the di¤erences of the formulas for particular case even they looks the same.
In the following table we present Cassini’s identities and Binet formulas for some Horadam (Fibonacci, Lucas, Pell, Jacobsthal) quaternions:
Table 5.
Horadam22-ions: Cassini’s Identity
c
Wm 1Wcm+1 Wcm2
Binet Formula
Fibonacci Quaternion ( 1)
m 1 1 p5 2 bb
1+p5 2 bb
p 5
b 1+p5 2
n
b 1 p5 2
n
p 5
Lucas Quaternion 5( 1)
m 1 1 p5 2 bb
1+p5 2 bb
p 5
p
5b 1+p5 2
n p
5b 1 2p5 n
Pell Quaternion ( 1)
m 1((1 p2)bb (1+p2)bb)
2p2
b(1+p2)n b (1 p2)n
2p2
Jacobsthal Quaternion ( 2)m 1(3bb 2bb) b2n b3( 1)n where b= 3 X s=0 se s; b=
3
X
s=0
se s:
In the next table we give Cassini’s identities and Binet formulas for some Horadam (Fibonacci, Lucas, Pell, Jacobsthal) octonions:
Table 6.
Horadam23-ions: Cassini’s Identity
c
Wm 1Wcm+1 cWm2
Binet Formula
Fibonacci Octonions ( 1)
m 1 1 p5 2 bb
1+p5 2 bb
p 5
b 1+p5 2
n
b 1 p5 2
n
p 5
Lucas Octonions 5( 1)
m 1 1 p5 2 bb
1+p5 2 bb
p 5
p
5b 1+p5 2
n p
5b 1 2p5 n
Pell Octonions ( 1)
m 1((1 p2)
bb (1+p2)bb)
2p2
b(1+p2)n b (1 p2)n
2p2
Jacobsthal Octonions ( 2)m 1(3bb 2bb) b2n b3( 1)n where b= 7 X s=0 se s; b=
7
X
s=0
se s:
In the following table we present Cassini’s identities and Binet formulas for some Horadam (Fibonacci, Lucas, Pell, Jacobsthal) sedenions:
Horadam24-ions: Cassini’s Identity
c
Wm 1Wcm+1 Wcm2
Binet Formula
Fibonacci Sedenions ( 1)
m 1 1 p5 2 bb
1+p5 2 bb
p 5
b 1+p5 2
n
b 1 p5 2
n
p 5
Lucas Sedenions 5( 1)
m 1 1 p5 2 bb
1+p5 2 bb
p 5
p
5b 1+p5 2
n p
5b 1 p5 2
n
Pell Sedenions ( 1)
m 1
((1 p2)bb (1+p2)bb)
2p2
b(1+p2)n b(1 p2)n
2p2
Jacobsthal Sedenions ( 2)m 1(3bb 2bb) b2n b3( 1)n where b= 15 X s=0 se s; b=
15
X
s=0
se s:
In the following table we give Cassini’s identities and Binet formulas for some Horadam (Fibonacci, Lucas, Pell, Jacobsthal) trigintaduonions:
Table 8.
Horadam25-ions: Cassini’s Identity
c
Wm 1cWm+1 cWm2
Binet Formula
Fibonacci Trigintaduonions ( 1)
m 1 1 p5 2 bb
1+p5 2 bb
p 5
b 1+p5 2
n
b 1 p5 2
n
p 5
Lucas Trigintaduonions 5( 1)
m 1 1 p5 2 bb
1+p5 2 bb
p 5
p
5b 1+p5 2
n p
5b 1 2p5 n
Pell Trigintaduonions ( 1)
m 1((1 p2)bb (1+p2)bb)
2p2
b(1+p2)n b (1 p2)n
2p2
Jacobsthal Trigintaduonions ( 2)m 1(3bb 2bb) b2n b3( 1)n where b= 31 X s=0 se s; b=
31
X
s=0
se s:
Remark 1. The formulas in the above last four tables looks the same but notice the di¤ erencies of b
andb’sfor each case ofk.
The following theorem gives d’Ocagne’s identity for Horadam2k-ions.
Theorem 5 (d’Ocagne’s Identity). For any integersmandn; we have
(3.14) WcmWcn+1 Wcm+1cWn=
AB( n m m n)( bb bb)
( )2 :
Proof. By using Binet formula, we obtain
c
WmWcn+1 Wcm+1cWn=
Ab m Bb m
Ab n+1 Bb n+1
Ab m+1 Bb m+1
Ab n Bb n
After necessary calculations we get the identity (3.14).
In the next Theorem, we give the summation formula for the …rstn+ 1 Horadam2k-ions.
Theorem 6. The sum formula for Horadam 2k-ions is
(3.15)
n
X
i=0
c Wi=
1 Bb n+1
1
Ab n+1
1
!
where
C=Ab(1 ) Bb(1 )
( ) (1 ) (1 ) :
Proof. By using the Binet formula and then geometric series we obtain
n
X
i=0
c Wi =
Ab Xn
i=0
n Bb n
X
i=0
n
= Ab 1
n+1
1
Bb 1 n+1
1 :
After some calculations we get
n
X
i=0
c Wi=
1 Bb n+1
1
Ab n+1
1
!
+Ab(1 ) Bb(1 )
( ) (1 ) (1 ) :
This completes the proof.
By using the initial values and roots of characteristic equation we can state the norm ofnth Horadam
2k-ions as follows.
Theorem 7. The norm ofnth Horadam 2k-ions is
(3.16) Nr(Wcn) =
A2 2n(1 + 2+ 4+ + N 2) +B2 2n
(1 + 2+ 4+ + N 2)
( )2 D
where
D= 2AB( q)
n(a+ ( q) + + ( q)N2 1)
( )2 andN = 2
k:
Proof. We have de…nednthHoradam 2k-ions as
c
Wn=Wne0+Wn+1e1+ +Wn+N 1eN 1:
Thus, norm ofnthHoradam2k-ions is
Nr(cWn) =cWnWcn=WcnWcn=Wn2+Wn2+1+ +Wn2+N 1:
Making necessary calculations and using the identities + =pand = q;we obtain
Nr(cWn) =
A2 2n(1 + 2+ 4+ + N 2)
( )2 +
B2 2n
(1 + 2+ 4+ + N 2)
( )2
2AB( q)n(a+ ( q) + + ( q)N2 1)
4. Matrix Methods in Horadam 2k-ions
We now consider the following two special cases offWng:
Un : =Un(p; q) =Wn(0;1;p; q);
Vn : =Vn(p; q) =Wn(2; p;p; q);
ThenfUngandfVngcan be expressed in the form
Un = ( n n)=( ); (4.1)
Vn = n+ n: (4.2)
These numbersUn andVn are called fundamental numbers and primordial numbers, respectively. Note that ifp= 1; q= 1then(Un)and(Vn)are classical Fibonacci and Lucas sequences, i.e,
Fn = Wn(0;1; 1;1) =Un;
Ln = Wn(2;1; 1;1) =Vn:
The matrix method is very useful method in order to obtain some identities for special sequences. We de…ne the square Horadam matrixM of order2as:
M =
0 @ p q
1 0
1 A=
0
@ U2 qU1
U1 qU0
1 A:
Note that
(4.3) Mn=
0
@ Un+1 qUn Un qUn 1
1 A:
For a proof of (4.3), see [8]. We can give a Corollary of the formula (4.3).
Corollary2. [8] For every n 1we have
(a): det(Mn) = ( q)n
(b): Un+1Un 1 Un2= ( q)n 1 (Cassini formula).
Writing the recurrence (2.1) as
0 @ Wn
Wn 1
1 A=
0 @ p q
1 0
1 A
0 @ Wn 1
Wn 2
1 A
in matrix form leads readily to the matrix power equation
(4.4)
0 @ Wn
Wn 1
1 A=
0 @ p q
1 0
1 A
n 10
@ W1
W0
which holds forn 1:Also we can write
0 @ Un+1
Un
1 A=
0 @ p q
1 0
1 A
0 @ Un
Un 1
1 A
and 0
@ Vn+1 Vn
1 A=
0 @ p q
1 0
1 A
0 @ Vn
Vn 1
1 A:
Now we de…ne the matrixMW as
(4.5) MW =
0
@ cW2 qcW1
c W1 qcW0
1 A:
This matriceMW can be called Horadam2k-ions matrix.
Theorem 8. Forn 0;the following holds:
(4.6) MW
0 @ p q
1 0 1 A n = 0
@ Wcn+2 qWcn+1 c
Wn+1 qWcn
1 A
Proof. We prove by mathematical induction onn:If n= 0then the result is clear. Now, we assume it is true forn=k; that is
MWMk=
0
@ cWk+2 qcWk+1 c
Wk+1 qWck
1 A:
If we use (3.2), then for k 1; we have cWk+3 = pWck+2+qcWk+1 and Wck+2 = pWck+1+qWck: Then by induction hypothesis, we obtain
MWMk+1 = (MWMk)M =
0
@ Wck+2 qWck+1 c
Wk+1 qcWk
1 A
0 @ p q
1 0
1 A
=
0
@ pcWk+2+qcWk+1 qcWk+2 pWck+1+qWck qcWk+1
1 A=
0
@ Wck+3 qWck+2 c
Wk+2 qWck+1
1 A
=
0
@ Wc(k+1)+2 qWc(k+1)+1
c
W(k+1)+1 qWc(k+1)
1 A
Thus, (4.6) holds for all non-negative integersn:
Corollary3. Forn 0; the following holds:
c
Wn+1=Wc1Un+1+qWc0Un:
Proof. The proof can be seen by using the matricies in (4.3) and (4.5) and the identity (4.6).
Corollary4. Forn 0; the following (Cassini formula) holds:
c
Proof. From (4.6), we have
det
0
@ cWn+2 qWcn+1
c
Wn+1 qcWn
1
A = det (MW) det
0 @ p q
1 0
1 A
n
) cWn+2qWcn Wcn+1qcWn+1= (qcW2cW0 qWc12)( q)n
) cWn+1Wcn 1 Wcn2= (Wc2Wc0 cW12)( q)n 1:
Theorem9. Letnandmnonnegative integers. Then we have the following relation between fundamental and primordial2k-ions, i.e.,Ubn andVbn,
(4.7) Vbn+m=Um+1Vbn+qUmVbn 1:
Proof. Ifm= 0then
b
Vn+0=U0+1Vbn+qU0Vbn 1
which is true. Assume that the equality holds form k: Form=k+ 1;we have
b
Vn+(k+1) = pVbn+k+qVbn+(k 1)
= p(Uk+1Vbn+qUkVbn 1) +q(U(k 1)+1Vbn+qU(k 1)Vbn 1)
= (pUk+1+qUk)Vbn+q(pUk+qU(k 1))Vbn 1
= Uk+2Vbn+qUk+1Vbn 1=U(k+1)+1Vbn+qU(k+1)Vbn 1:
By strong induction onm; this proves (4.7).
Second method. We use Theorem 1 and formula (4.1). Note that A = b a = p 2 and B =
b a =p 2 :Now we have
Um+1Vbn+qUmVbn 1 =
m+1 m+1(p 2 )b n (p 2 )b n
+q
m m
(p 2 )b n 1 (p 2 )b n 1
= (p 2 )b
n+m (p 2 )b n+m
= Vbn+m:
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Zonguldak Bulent Ecevit University, Department of Mathematics, Zonguldak, Turkey.
