Paper-22-20-2009. 98KB Jun 04 2011 12:03:12 AM
ACTA UNIVERSITATIS APULENSIS
No 20/2009
ON THE FIBONACCI Q-MATRICES OF THE ORDER m
Serpil Halıcı, Tuna Batu
Abstract. In this paper, the Fibonacci Q-matrices of the order m and their
properties are considered. In addition to this, we introduce the Fibonacci Q-matrices
of the order a negative real number m and the pure imaginary m. Further, we
examine some properties of these matrices.
2000 Mathematics Subject Classification: 11B37, 11B39,11B50.
1. Introduction
In the last decades the theory of Fibonacci numbers was complemented by the
theory of the so-called Fibonacci Q-matrix (see [1], [2], [5]). This 2×2 square matrix
is defined in [5] as follows.
#
"
1 1
.
(1)
Q=
1 0
It is well known that, (see [7]), the nth power of the Q-matrix is
n
Q =
"
Fn+1 Fn
Fn Fn−1
#
(2)
,
where n = 0, ±1, ±2, · · ·, and Fn is n th Fibonacci number. Qn matrix is expressed
by the formula
n
Q =
"
Fn + Fn−1 Fn−1 + Fn−2
Fn−1 + Fn−2 Fn−2 + Fn−3
#
=
"
Fn Fn−1
Fn−1 Fn−2
#
+
"
Fn−1 Fn−2
Fn−2 Fn−3
#
.
The Fibonacci Q-matrices of the order m are considered by Stakhov in [1] . These
matrices are defined by Stakhov as follows, for m ∈ R+
Gm =
"
m 1
1 0
211
#
.
(3)
S. Halıcı, T. Batu - On the Fibonacci Q-matrices of the order m
It is well known that the Fibonacci numbers of order m is defined by the following
recurrence relation for n ∈ Z,
Fm (n + 2) = mFm (n + 1) + Fm (n)
(4)
where Fm (0) = 0 and Fm (1) = 1. The entries of these matrices of order m can be
represented by Fibonacci numbers of order m as follows,
Gnm
=
"
Fm (n + 1)
Fm (n)
Fm (n)
Fm (n − 1)
#
(5)
,
where m ∈ R+ and n ∈ Z. If we consider the determinants of the Gnm matrices of the
order m, then we can see that
Det (Gnm )
F (n + 1)
Fm (n)
= m
Fm (n)
Fm (n − 1)
n
= (−1) .
(6)
It can be seen that the Cassini Formula for the Fibonacci numbers of the order
m is
Fm (n + 1) Fm (n − 1) − [Fm (n)]2 = (−1)n .
(7)
Theorem 1. For m ∈ R+ and n ∈ Z,
n−2
Gnm = mGn−1
m + Gm .
(8)
Proof. Since,
Gn−1
=
m
"
Fm (n)
Fm (n − 1)
Fm (n − 1) Fm (n − 2)
#
(9)
Gn−2
m
"
Fm (n − 1) Fm (n − 2)
Fm (n − 2) Fm (n − 3)
#
(10)
and
=
,
one can write
n−2
n−1
mGm
+ Gm
=
mFm (n) + Fm (n − 1)
mFm (n − 1) + Fm (n − 2)
mFm (n − 1) + Fm (n − 2) mFm (n − 2) + Fm (n − 3)
"
#
n−2
mGn−1
=
m + Gm
Fm (n + 1)
Fm (n)
Fm (n)
Fm (n − 1)
Thus, the proof is completed.
212
.
(11)
(12)
S. Halıcı, T. Batu - On the Fibonacci Q-matrices of the order m
2. The Fibonacci Q-Matrices of the Order m
For m ∈ R+ , the Fibonacci Q-matrices of order −m can be defined by
G−m =
"
−m 1
1 0
#
(13)
.
Since, the Fibonacci numbers of order −m can be defined by the following
recurrence relation , for m ∈ R+ , n ∈ Z
F−m (n + 2) = −mF−m (n + 1) + F−m (n)
(14)
where F−m (0) = 0, F−m (1) = 1.The following theorem sets a connection of the
G−m matrix with the generalized Fibonacci numbers of order −m.
Theorem 2.For m ∈ R+ and n ∈ Z,
Gn−m =
"
F−m (n + 1)
F−m (n)
F−m (n)
F−m (n − 1)
#
.
(15)
Proof. Since, F−m (0) = 0, F−m (1) = 1 and
F−m (2) = −m,
F−m (3) = m2 + 1,
F−m (4) = −m3 − 2m,
F−m (5) = m4 + 3m2 + 1, ...
we can write
#
"
# "
−m
1
F
(2)
F
(1)
−m
−m
G1−m =
,
=
F−m (1) F−m (0)
1 0
G2−m =
−m 1
1
0
G3−m
=
−m 1
1
0
=
−m3 − 2m
m2 + 1
m2 + 1 −m
F−m (3) F−m (2)
=
,
−m
1
F−m (2) F−m (1)
F−m (5) F−m (4)
m2 + 1
.
=
F−m (4) F−m (3)
−m
(16)
(17)
(18)
By the inductive method, we can easily verify that
Gn−m
=
F−m (n + 1)
F−m (n)
F−m (n)
F−m (n − 1)
.
(19)
Theorem 3. For a given integer n and a positive real m number we have that
213
S. Halıcı, T. Batu - On the Fibonacci Q-matrices of the order m
Det Gn−m = (−1)n .
(20)
Proof. Using general properties of the determinants the proof can be easily seen.
It is clear that the above equation is a generalized of the famous Cassini formula.
Theorem 4. For a given integer n and a positive real number m we have that
n−2
Gn−m = −mGn−1
−m + G−m .
(21)
Proof. From the recursive relation F−m (n + 2) = −mF−m (n + 1) + F−m (n) we
can write
F−m (n − 1) F−m (n − 2)
,
+
F−m (n − 2) F−m (n − 3)
(22)
−mF−m (n) + F−m (n − 1)
−mF−m (n − 1) + F−m (n − 2)
=
= Gn−m .
−mF−m (n − 1) + F−m (n − 2) −mF−m (n − 2) + F−m (n − 3)
(23)
n−1
n−2
+G−m
=
−mG−m
n−2
−mGn−1
−m +G−m
−mF−m (n)
−mF−m (n − 1)
−mF−m (n − 1) −mF−m (n − 2)
3. The Fibonacci Q-Matrices of the Order
√
−m
√
The Fibonacci Q-matrices with the order −m can be defined by the following
expression
" √
#
−m
1
,
(24)
G√−m =
1
0
√
where m is a positive number. The Fibonacci numbers with order −m is defined
by the following recurrence relation,
√
(25)
F√−m (n + 2) = −mF√−m (n + 1) + F√−m (n) ,
F√−m (0) = 0, F√−m (1) = 1 and m ∈ R+ , n ∈ Z.
Theorem 5. For a given integer n and a positive real m numberwe have that
n
G√−m =
"
F√−m (n + 1)
F√−m (n)
F√−m (n − 1)
F√−m (n)
214
#
.
(26)
S. Halıcı, T. Batu - On the Fibonacci Q-matrices of the order m
Proof. Since, F√−m (2) =
√
2 mi, · · · one can write
G1√
−m
=
√
√
mi , F√−m (3) = −m + 1 , F√−m (4) = −m mi +
" √
mi 1
1
0
#
=
"
F√−m (2) F√−m (1)
F√−m (1) F√−m (0)
#
(27)
and
G4√
−m
=
#
" √
√
√
2
F −m (5) F√−m (4)
m√
− 3m +√1
−m mi + 2 mi
=
.
F√−m (4) F√−m (3)
−m mi + 2 mi
−m + 1
(28)
Thus, by the inductive method we obtain that
n
G√−m =
"
F√−m (n)
F√−m (n + 1)
F√−m (n − 1)
F√−m (n)
#
.
(29)
Taking into account the Cassini formula we can write
Det Gn√−m = (−1)n .
Thus, we can also write
h
(30)
i2
F√−m (n + 1) F√−m (n − 1) − F√−m (n)
= (−1)n .
(31)
This
equation can be called Cassini Formula of Fibonacci numbers of the order
√
−m.
√
Using the recursive relation F√−m (n + 2) = −mF√−m (n + 1) + F√−m (n) we
write the following theorem.
Theorem 6.
Gn√−m =
√
√
√
+ Gn−2
.
−mGn−1
−m
−m
(32)
Proof. The proof is immediately follows from the definition Gn√−m .
Aknowledgement. Research of the first author is supported by Sakarya University, Scientific Research Project Unit.
215
S. Halıcı, T. Batu - On the Fibonacci Q-matrices of the order m
References
[1] A. P. Stakhov, The golden matrices and a new kind of cryptography, Chaos,
Solitions and Fractals 32(2007), 1138-1146.
[2] A. Stakhov and B. Rozin, The golden shofar, Chaos, Solitions and Fractals,
26 (2005), 677-684.
[3] A. P. Stakhov, Fibonacci matrices, a generalization of the Cassini formula
and a new coding theory, Solitions and Fractals, 30 (2006), 56-66.
[4] http://www.goldenmuseum.com/0905 HyperFibLuck engl.html
[5] T. Koshy, Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience
publication, U.S.A, (2001).
[6] F. E. Hohn,Elementary Matrix Algebra, Macmillan Company, New York,
(1973).
[7] V. E. Hoggat, Fibonacci and Lucas numbers, Houghton-Mifflin, Palo Alto,
(1969).
Serpil Halici
Department of Mathematics
Faculty of Arts and Sciences
University of Sakarya
Turkey.
email: shalici@sakarya.edu.tr
216
No 20/2009
ON THE FIBONACCI Q-MATRICES OF THE ORDER m
Serpil Halıcı, Tuna Batu
Abstract. In this paper, the Fibonacci Q-matrices of the order m and their
properties are considered. In addition to this, we introduce the Fibonacci Q-matrices
of the order a negative real number m and the pure imaginary m. Further, we
examine some properties of these matrices.
2000 Mathematics Subject Classification: 11B37, 11B39,11B50.
1. Introduction
In the last decades the theory of Fibonacci numbers was complemented by the
theory of the so-called Fibonacci Q-matrix (see [1], [2], [5]). This 2×2 square matrix
is defined in [5] as follows.
#
"
1 1
.
(1)
Q=
1 0
It is well known that, (see [7]), the nth power of the Q-matrix is
n
Q =
"
Fn+1 Fn
Fn Fn−1
#
(2)
,
where n = 0, ±1, ±2, · · ·, and Fn is n th Fibonacci number. Qn matrix is expressed
by the formula
n
Q =
"
Fn + Fn−1 Fn−1 + Fn−2
Fn−1 + Fn−2 Fn−2 + Fn−3
#
=
"
Fn Fn−1
Fn−1 Fn−2
#
+
"
Fn−1 Fn−2
Fn−2 Fn−3
#
.
The Fibonacci Q-matrices of the order m are considered by Stakhov in [1] . These
matrices are defined by Stakhov as follows, for m ∈ R+
Gm =
"
m 1
1 0
211
#
.
(3)
S. Halıcı, T. Batu - On the Fibonacci Q-matrices of the order m
It is well known that the Fibonacci numbers of order m is defined by the following
recurrence relation for n ∈ Z,
Fm (n + 2) = mFm (n + 1) + Fm (n)
(4)
where Fm (0) = 0 and Fm (1) = 1. The entries of these matrices of order m can be
represented by Fibonacci numbers of order m as follows,
Gnm
=
"
Fm (n + 1)
Fm (n)
Fm (n)
Fm (n − 1)
#
(5)
,
where m ∈ R+ and n ∈ Z. If we consider the determinants of the Gnm matrices of the
order m, then we can see that
Det (Gnm )
F (n + 1)
Fm (n)
= m
Fm (n)
Fm (n − 1)
n
= (−1) .
(6)
It can be seen that the Cassini Formula for the Fibonacci numbers of the order
m is
Fm (n + 1) Fm (n − 1) − [Fm (n)]2 = (−1)n .
(7)
Theorem 1. For m ∈ R+ and n ∈ Z,
n−2
Gnm = mGn−1
m + Gm .
(8)
Proof. Since,
Gn−1
=
m
"
Fm (n)
Fm (n − 1)
Fm (n − 1) Fm (n − 2)
#
(9)
Gn−2
m
"
Fm (n − 1) Fm (n − 2)
Fm (n − 2) Fm (n − 3)
#
(10)
and
=
,
one can write
n−2
n−1
mGm
+ Gm
=
mFm (n) + Fm (n − 1)
mFm (n − 1) + Fm (n − 2)
mFm (n − 1) + Fm (n − 2) mFm (n − 2) + Fm (n − 3)
"
#
n−2
mGn−1
=
m + Gm
Fm (n + 1)
Fm (n)
Fm (n)
Fm (n − 1)
Thus, the proof is completed.
212
.
(11)
(12)
S. Halıcı, T. Batu - On the Fibonacci Q-matrices of the order m
2. The Fibonacci Q-Matrices of the Order m
For m ∈ R+ , the Fibonacci Q-matrices of order −m can be defined by
G−m =
"
−m 1
1 0
#
(13)
.
Since, the Fibonacci numbers of order −m can be defined by the following
recurrence relation , for m ∈ R+ , n ∈ Z
F−m (n + 2) = −mF−m (n + 1) + F−m (n)
(14)
where F−m (0) = 0, F−m (1) = 1.The following theorem sets a connection of the
G−m matrix with the generalized Fibonacci numbers of order −m.
Theorem 2.For m ∈ R+ and n ∈ Z,
Gn−m =
"
F−m (n + 1)
F−m (n)
F−m (n)
F−m (n − 1)
#
.
(15)
Proof. Since, F−m (0) = 0, F−m (1) = 1 and
F−m (2) = −m,
F−m (3) = m2 + 1,
F−m (4) = −m3 − 2m,
F−m (5) = m4 + 3m2 + 1, ...
we can write
#
"
# "
−m
1
F
(2)
F
(1)
−m
−m
G1−m =
,
=
F−m (1) F−m (0)
1 0
G2−m =
−m 1
1
0
G3−m
=
−m 1
1
0
=
−m3 − 2m
m2 + 1
m2 + 1 −m
F−m (3) F−m (2)
=
,
−m
1
F−m (2) F−m (1)
F−m (5) F−m (4)
m2 + 1
.
=
F−m (4) F−m (3)
−m
(16)
(17)
(18)
By the inductive method, we can easily verify that
Gn−m
=
F−m (n + 1)
F−m (n)
F−m (n)
F−m (n − 1)
.
(19)
Theorem 3. For a given integer n and a positive real m number we have that
213
S. Halıcı, T. Batu - On the Fibonacci Q-matrices of the order m
Det Gn−m = (−1)n .
(20)
Proof. Using general properties of the determinants the proof can be easily seen.
It is clear that the above equation is a generalized of the famous Cassini formula.
Theorem 4. For a given integer n and a positive real number m we have that
n−2
Gn−m = −mGn−1
−m + G−m .
(21)
Proof. From the recursive relation F−m (n + 2) = −mF−m (n + 1) + F−m (n) we
can write
F−m (n − 1) F−m (n − 2)
,
+
F−m (n − 2) F−m (n − 3)
(22)
−mF−m (n) + F−m (n − 1)
−mF−m (n − 1) + F−m (n − 2)
=
= Gn−m .
−mF−m (n − 1) + F−m (n − 2) −mF−m (n − 2) + F−m (n − 3)
(23)
n−1
n−2
+G−m
=
−mG−m
n−2
−mGn−1
−m +G−m
−mF−m (n)
−mF−m (n − 1)
−mF−m (n − 1) −mF−m (n − 2)
3. The Fibonacci Q-Matrices of the Order
√
−m
√
The Fibonacci Q-matrices with the order −m can be defined by the following
expression
" √
#
−m
1
,
(24)
G√−m =
1
0
√
where m is a positive number. The Fibonacci numbers with order −m is defined
by the following recurrence relation,
√
(25)
F√−m (n + 2) = −mF√−m (n + 1) + F√−m (n) ,
F√−m (0) = 0, F√−m (1) = 1 and m ∈ R+ , n ∈ Z.
Theorem 5. For a given integer n and a positive real m numberwe have that
n
G√−m =
"
F√−m (n + 1)
F√−m (n)
F√−m (n − 1)
F√−m (n)
214
#
.
(26)
S. Halıcı, T. Batu - On the Fibonacci Q-matrices of the order m
Proof. Since, F√−m (2) =
√
2 mi, · · · one can write
G1√
−m
=
√
√
mi , F√−m (3) = −m + 1 , F√−m (4) = −m mi +
" √
mi 1
1
0
#
=
"
F√−m (2) F√−m (1)
F√−m (1) F√−m (0)
#
(27)
and
G4√
−m
=
#
" √
√
√
2
F −m (5) F√−m (4)
m√
− 3m +√1
−m mi + 2 mi
=
.
F√−m (4) F√−m (3)
−m mi + 2 mi
−m + 1
(28)
Thus, by the inductive method we obtain that
n
G√−m =
"
F√−m (n)
F√−m (n + 1)
F√−m (n − 1)
F√−m (n)
#
.
(29)
Taking into account the Cassini formula we can write
Det Gn√−m = (−1)n .
Thus, we can also write
h
(30)
i2
F√−m (n + 1) F√−m (n − 1) − F√−m (n)
= (−1)n .
(31)
This
equation can be called Cassini Formula of Fibonacci numbers of the order
√
−m.
√
Using the recursive relation F√−m (n + 2) = −mF√−m (n + 1) + F√−m (n) we
write the following theorem.
Theorem 6.
Gn√−m =
√
√
√
+ Gn−2
.
−mGn−1
−m
−m
(32)
Proof. The proof is immediately follows from the definition Gn√−m .
Aknowledgement. Research of the first author is supported by Sakarya University, Scientific Research Project Unit.
215
S. Halıcı, T. Batu - On the Fibonacci Q-matrices of the order m
References
[1] A. P. Stakhov, The golden matrices and a new kind of cryptography, Chaos,
Solitions and Fractals 32(2007), 1138-1146.
[2] A. Stakhov and B. Rozin, The golden shofar, Chaos, Solitions and Fractals,
26 (2005), 677-684.
[3] A. P. Stakhov, Fibonacci matrices, a generalization of the Cassini formula
and a new coding theory, Solitions and Fractals, 30 (2006), 56-66.
[4] http://www.goldenmuseum.com/0905 HyperFibLuck engl.html
[5] T. Koshy, Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience
publication, U.S.A, (2001).
[6] F. E. Hohn,Elementary Matrix Algebra, Macmillan Company, New York,
(1973).
[7] V. E. Hoggat, Fibonacci and Lucas numbers, Houghton-Mifflin, Palo Alto,
(1969).
Serpil Halici
Department of Mathematics
Faculty of Arts and Sciences
University of Sakarya
Turkey.
email: shalici@sakarya.edu.tr
216