Some Identities for an Alternating Sum of Fibonacci
and Lucus Numbers of Order k
SUKANYA SOMPROM, WAITAYA NIMNUAL, WATHCHARAPONG HONGTHONG
Department of Mathematics,
Faculty of Science and Technology,
Surindra Rajabhat University, Surin 32000,
THAILAND
Abstract: - In this paper, we defined
)(k
n
F
be the Fibonacci of order
k
and
)(k
n
L
be the Lucas number of order
k
. We presented some of their new identities as well as some results of relation for an alternating sum between
Fibonacci and Lucas number of order
k
as follow;
)(
2
)(
1
0
142)1( k
i
k
i
n
i
ni FmmLm
k
j
kji
k
ijFmFm
3
)(
2
)(
1
2
2
)()(
12)1( k
n
k
n
nFFm
.
Key-Words: - Alternating sums, Fibonacci number of order
k
, Lucas number of order
k
, Tribonacci number
Received: October 12, 2021. Revised: May 29, 2022. Accepted: July 2, 2022. Published: July 20, 2022.
1 Introduction
In recent years, the Fibonacci and Lucas number are
an ordinal number that is significant in mathematics.
It has also gained widespread attention in applied
mathematics, number theory, computers, etc. It can
also be applied in other fields such as art,
architecture, finance, etc. Moreover, it is also applied
in financial. Investors use it to look at the price
reversal of the asset, for example. In the application
of art and architecture, the Golden Ratio of the
Fibonacci sequence which is the theory that
calculates the most beautiful proportions in the world
is used in urban design. Even the Mona Lisa painting
also uses the Golden Ratio. In nature, Fibonacci can
also explain the spiral rotation of sunflower seeds or
a cactus with a thorny arrangement corresponding to
the Fibonacci sequence. The Fibonacci sequence was
first presented in 1202 by Leonardo Fibonacci.
Let
n
F
be the Fibonacci number defined
recursively as follows
21 nnn FFF
,
for
2n
with
0
0F
and
1
1F
. For example, the
sequence of the Fibonacci numbers are 0, 1, 1, 2, 3,
5, 8, 13, 21, … .
Let
n
L
be the Lucas number defined recursively
as follows
21 nnn LLL
,
for
2n
with
2
0L
and
1
1L
. For example, the
sequence of the Lucas numbers are 2, 1, 3, 4, 7, 11,
18, 29, 47, … .
Researchers have been talking about the
properties of Fibonacci and Lucas number for a long
time. In 2016, Tom Edgar [1] produced some
properties of an alternating sum Fibonacci-Lucas
relations as demonstrated below:
1
0
1
0
1
1)2(
m
ii
i
m
ii
i
m
mFkkLkFk
,
for
km,
. In addition, there were also researchers
who studied about alternating sum, Zvonko Cerin
studied formulas the Lucas number for the odd and
even terms and Emrah Kilic et al. showed some
results of the Fibonacci and Lucas numbers from
alternating Melham’s sums, see [2, 3] for details.
Let
k
n
F
be the
k
-Fibonacci number defined
recursively as follows
k
n
k
n
k
nFkFF11
,
for
1n
with
0
0
k
F
and
1
1
k
F
.
Let
k
n
L
be the
k
-Lucas number defined
recursively as follows
k
n
k
n
k
nLkLL11
,
for
1n
with
2
0
k
L
and
1
1
k
L
.
The
k
-Fibonacci and
k
-Lucas numbers have
been introduced on properties such as Yashwant K.
Panwar et al. [4] studied the sums of odd and even
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.65
Sukanya Somprom,
Waitaya Nimnual, Wathcharapong Hongthong
E-ISSN: 2224-2880
580
Volume 21, 2022
terms of the
k
-Fibonacci numbers by using the
Binet’s formula. In 2014, Bijindra Singh et al. [5]
studied the products of
k
-Fibonacci and
k
-Lucas
numbers using the Binet’s formulas. Moreover,
There are people who are interested in studying the
relationship of
k
-Fibonacci and
k
-Lucas numbers
in various forms, see [6, 7] for details. In addition,
there are still many researchers who are interested in
studying and developing the
),( qp
- Fibonacci and
),( qp
-Lucas number. For example, in 2017,
Alongkot S. and Mongkol T. [8] studied the product
of
),( qp
- Fibonacci and
),( qp
-Lucas number by
using the Binet’ formulas to show their properties.
2 Preliminaries
In section 2, we introduced the fundamental
Definitions of the Fibonacci and Lucas number of
order
k
and Corollary.
Definition 1 For
)1(, kkn
, let
)(k
n
F
be the
Fibonacci number of order
k
defined recursively as
follows:
k
i
kjn
k
nFF
1
)()(
,
for
2n
with
0
)(
k
n
F
for
01 nk
and
1
)(
1
k
F
. In case of
2k
the equation is as follows:
)2(
2
)2(
1
)2(
nnn FFF
or
21 nnn FFF
,
that is, the equation of the Fibonacci sequence.
k \ n
1
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
4
4
4
4
4
4
5
7
8
8
8
8
8
8
13
15
16
16
16
16
13
24
29
31
32
32
32
21
44
56
61
63
64
64
34
81
108
120
125
127
128
Table 1 Fibonacci numbers of order
k
Definition 2 For
)1(, kkn
, let
)( k
n
L
be the
Lucas number of order
k
defined recursively as
follows:
1
1
)()( n
j
kjn
k
nLnL
,
for
kn 2
and
k
j
kjn
k
nLL
1
)()(
,
for
1 kn
with
kLk
)(
0
and
1
)(
1
k
L
. In case of
2k
the equation is as follows:
)2(
2
)2(
1
)2(
nnn LLL
or
21 nnn LLL
,
that is, the equation of the Lucas sequence.
k \ n
1
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
1
1
1
1
1
1
1
3
3
3
3
3
3
3
4
7
7
7
7
7
7
7
11
15
15
15
15
15
11
21
26
31
31
31
31
18
39
51
57
63
63
63
29
71
99
113
120
127
127
47
131
191
223
239
247
255
76
241
367
439
475
493
502
Table 2 Lucas number of order
k
Corollary 1 [9] The Lucas number of order
k
are
expressed in terms of the Fibonacci numbers of order
k by
},min{
1
)(
1
)(
kn
j
kjn
k
njFL
, (1)
for
)1(, kkn
.
A number of researchers have studied the
Fibonacci and Lucas numbers of order
k
using
Corollary 1 in proving various Theorem and
Lemmas, see for details [10].
In 2020, Spios D. Dafnis et al. [11] introduced
some identities of Fibonacci and Lucas number of
order k as follow:
n
i
k
j
kji
k
i
k
i
ini jFFmLm
0 3
)(
2
)()(
1)2()1(
)(
1
)1( k
n
nF
.
In this study, we presented some identities of the
Fibonacci and Lucas number of order
k
defined by
Definition 1, Definition 2, and the Corollary to show
the theorem.
3 Main Theorem
In this section, we gave some theorem of the
Fibonacci and Lucas number of order
k
. First, we
provided a lemma used in the proofs of our results.
Additionally, we presented corollary and example as
follows.
Lemma 1 Let
)(k
n
F
and
)( k
n
L
be the Fibonacci and
Lucas number of order
k
, respectively, then
},min{
3
)(
2
)()(
1
)(
12
kn
j
kjn
k
n
k
n
k
njFFFL
.
Proof The result is obvious with deriving the
following by Corollary 1.
Then we have the following Theorem.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.65
Sukanya Somprom,
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Theorem 1 Let
0
)(
n
k
n
F
and
0
)(
n
k
n
L
be the
Fibonacci and Lucas number of order
k
. Then,
)(
2
)(
1
0
142)1( k
i
k
i
n
i
ni FmmLm
k
j
kji
k
ijFmFm
3
)(
2
)(
1
2
2
)()(
12)1( k
n
k
n
nFFm
. (2)
Proof By Lemma 1, consider
)(
2
)(
1
0
142)1( k
i
k
i
n
i
ni FmmLm
k
j
kji
k
ijFmFm
3
)(
2
)(
1
2
2
k
j
kji
k
i
k
i
n
i
ini jFmmFmFm
3
)(
2
)()(
1
0
2)1(
k
j
kji
k
i
k
i
k
ijFmFmFFm
3
)(
2
)(
1
2)()(
2242
)()(
1
0
2)1( k
i
k
i
n
i
ini mFmFm
)(
1
2)()(2 2444 k
i
k
i
k
iFmFFmm
)(
1
2)()(2)(
1
0
22)1( k
i
k
i
k
i
k
i
n
i
ini FmmFFmmFm
)(
1
kn mFm
)(
0
2)(
1
)(
1
2)(
2
11 22)1( kkkkn FmmFFmmFm
)(
1
2)(
2
)(
2
2)(
3
22 22)1( kkkkn FmmFFmmFm
)(
2
2)(
3
)(
3
2)(
4
33 22)1( kkkkn FmmFFmmFm
)(
1
2)()(2)(
122)1( k
n
k
n
k
n
k
n
nnn FmmFFmmFm
)()(
1211 k
n
n
k
n
nmFmF
)()(
121 k
n
k
n
nFFm
.
Thus, the proof is complete.
From equation (2) , we considered the Table 1
and Table 2, by choosing
7k
and
4n
as an
example. It was found that on the left side of equation
(2), it could be calculated as
m4
. And on the right,
it could be calculated as
m4
where both sides were
equal. From the above theorems, we obtained the
well-known identities for Fibonacci and Lucas of
order
.k
For
2k
, i.e.
nn FF
)2(
,
nn LL
)2(
,
Theorem 1 reduces to new identities in the following
corollary.
Corollary 2 Let
n
F
and
n
L
be the Fibonacci and
Lucas number, respectively, then
1
2
2
1
0
1242)1(
iii
n
i
ni FmFmmLm
nn
nFFm 2)1( 1
. (3)
Moreover,
11
0
18422)1(
iii
n
i
ni FFL
nn
nFF 22)1( 1
, (4)
for
2m
.
Proof The proof is similar to Theorem 1 by fixing
2k
in equation (2) and fixed
2m
in equation
(3), which implies Corollary 2.
From corollary 2 , we considered the Table 1
and Table, by choosing
5n
as an example. It was
found that on the left side of equation (3), it could be
calculated as
m2
. On the right side, it could be
calculated as
m2
. It was found that both were equal.
For
5,2 nm
, it was found that on the left side of
equation (4), it could be calculated as
4
. On the right
side, it could be calculated as
4
. It was found that
both were equal.
Let
n
T
be the sequence of Tribonacci number
defined recursively as follows
321 nnnn TTTT
for
3n
with
0
0T
,
1
1T
and
1
2T
. For
example, the sequence of the Tribonacci number are
0, 1, 1, 2, 4, 7, 13, 24, 44, … .
Let
n
C
be the sequence of Lucas number of
order 3 defined recursively as follows
321 nnnn CCCC
for
3n
with
3
0C
,
1
1C
and
3
2C
. For
example, the sequence of the Lucas number of order
3 are 3, 1, 3, 7, 11, 21, 39, 71, 131, … .
Considering Theorem 1, we were able to find the
relationship between Tribonacci number and Lucas
number of order 3.
Lemma 2 For
n
, let
n
T
be the sequence of
Tribonacci number and
n
C
be the sequence of Lucas
number of order 3, then we have
111 32 nnnn mTmTmTmC
,
for
0m
.
Proof We used Corollary 1 and fixed
3k
, then the
proof is completes.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.65
Sukanya Somprom,
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Corollary 3 For
n
, let
n
T
be the sequence of
Tribonacci number and
n
C
be the sequence of Lucas
number of order 3, then
1
2
0
2
13242)1(
iii
n
ii
ini mTTmTmmCm
nn
nTTm 2)1( 1
. (5)
Moreover,
nn
n
n
iiii
ni TTTTC 22)1(61222)1( 1
0
11
1
,
for
2m
.
Proof By Lemma 2, consider
1
2
0
2
13242)1(
iii
n
ii
ini mTTmTmmCm
n
iiii
ni mTmTmTm
0
11
132)1(
11
2
23242 iii mTTmTm
n
iiiii
ni TmmTTmmTm
0
1
22
1
122)1(
1
2
00
2
1
022)1(
TmmTTmmTmn
0
2
11
2
2
11 22)1( TmmTTmmTmn
1
2
22
2
3
22 22)1( TmmTTmmTmn
1
22
1
022)1( nnnn
nTmmTTmmTm
n
n
n
nmTmT 2)1()1( 1
nn
nTTm 2)1( 1
.
Then, the proof is complete in equation (5). Next we
fixed
2m
in equation (5), then
1
2
0
2
1)2(3)2(242222)1(
iii
n
ii
ini TTTC
n
iiii
ini TTC
0
11 61222)1(
.
By Lemmma 2, we have
111 6422 nnnn TTTC
.
Therefore,
nn
n
n
iiii
ni TTTTC 22)1(61222)1( 1
0
11
1
.
From corollary 3, we considered the Table 1 and
Table 2, by choosing
3m
and
5n
as an example.
It was found that on the left side of equation (5) it
could be calculated as 3. And on the right side, it
could be calculated as 3. It was found that both were
equal.
4 Conclusion
In this study, we considered the Fibonacci and Lucas
number of order
k
by using the definitions in section
2 to show that some identities concerning an
alternating sum of the Fibonacci and Lucas number
of order
k
. From the theorem we have studied, we
found that in case of
2k
, the relationship of
identity is in the form of alternating sum of the
Fibonacci and Lucas number. And in case of
3k
,
alternating sum of the Tribonacci and Lucas number
of order k is the theorem that we study covering
identity in the form of
3,2k
. And other cases in the
form presented, the result can be obtained according
to the theorem we studied. Moreover, we presented
Lemmas, corollaries and examples. For those who
are interested, relationships and sequence properties
can be found in the form of an alternating sum.
References:
[1] Tom Edgar, Extending some Fibonacci-Lucas
relations, Fibonacci Q, Vol.54, 2016, pp. 79.
[2] Zvonko Cerin, Some Alternating sums of Lucas
numbers, Central European Journal of
Mathematics, Vol.3, No.1, 2005, pp. 1-13.
[3] Emrah Kilic, Nese Omur and Yucel Turker
Ulutas, Alternating Sums of the Power of
Fibonacci and Lucas Numbers, Miskolc
mathematical Notes, Vol.12, No.1, 2011, pp. 87-
103.
[4] Yashwant K. Panwar, G. P. S. Rathore and
Richa Chawla, On Sums of Odd and Even
Terms of the k-Fibonacci Numbers, Global
Journal of Mathematical Analysis, Vol.2, No.3,
2014, pp. 115-119.
[5] Bijendra Singh, Kiran Sisodiya and Farooq
Ahmad, On the Products of 𝑘-Fibonacci
Numbers and 𝑘-Lucas Numbers, International
Journal of Mathematics and Mathematical
Sciences , Vol.21, 2014, pp. 1-4.
[6] Yasemin Tasyurdu, Nurdan Cobanoglu and
Zulkuf Dilmen, On the a New Family of k-
Fibonacci Numbers, Journal of Science and
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[7] Merve Tastan, Engin Ozkan and Anthony G.
Shannon, The generalized k-Fibonacci
polynomails and generalized k-Lucas
polynomail, Notes on Number Theory and
Discrete Mathematics, Vol.27, No.2, 2021, pp.
148-158.
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DOI: 10.37394/23206.2022.21.65
Sukanya Somprom,
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E-ISSN: 2224-2880
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Volume 21, 2022
[8] Alongkot S. and Mongkol T. , Some properties
of the product of
),( qp
- Fibonacci and
),( qp
- Lucas number, International Journal of
GEOMATE, Vol.13, 2017, pp. 16-19.
[9] Ch. A. Charalambides, Lucas numbers and
polynomials of order k and the length of the
longest circular success run, Fibonacci Q,
Vol.29, 1991, pp. 290-297.
[10] Andreas N. Fhilippou and Spiros D. dafnis, A
Simple proof of an Identity Generalizing
Fibonacci-Lucas Identities, Fibonacci Q, Vol.8,
2018, pp. 334-336.
[11] Spiros D.Dafnis, Andreas N.Philippou and
Ioannis E.Livieris, An Alternating Sum of
Fibonacci and Lucas Numbers of Order k,
Mathematics, MDPI, Vol.8, No.9, 2020, pp. 1-
4.
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DOI: 10.37394/23206.2022.21.65
Sukanya Somprom,
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E-ISSN: 2224-2880
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