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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 284261, 11 pages
doi:10.1155/2011/284261
Research Article
Some Finite Sums Involving Generalized Fibonacci
and Lucas Numbers
E. Kılıç,1 N. Ömür,2 and Y. T. Ulutaş2
1
Mathematics Department, TOBB University of Economics and Technology, Sogutozu,
06560 Ankara, Turkey
2
Mathematics Department, Kocaeli University, 41380 İzmit Kocaeli, Turkey
Correspondence should be addressed to N. Ömür, neseomur@kocaeli.edu.tr
Received 11 August 2011; Accepted 7 September 2011
Academic Editor: Gerald Teschl
Copyright q 2011 E. Kılıç et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
By considering Melham’s sums Melham, 2004, we compute various
more
general nonalternating
sums, alternating sums, and sums that alternate according to −1
Fibonacci and Lucas numbers.
n1
2
involving the generalized
1. Introduction
Let a, b, and p be assumed to be arbitrary nonzero complex numbers with pp2 2p2 4 / 0.
Define second-order linear recursion {Wn } by
Wn pWn−1 Wn−2,
1.1
with W0 a, W1 b for all integers n. Since Δ p2 4 /
0, the roots α and β of x2 − px − 1 0
are distinct.
Also define the sequence {Xn } via the terms of sequence {Wn } as Xn Wn1 Wn−1 .
The Binet formulas for the sequences {Wn } and {Xn } are
Wn Aαn − Bβn
,
α−β
where A b − aβ and B b − aα.
Xn Aαn Bβn ,
1.2
2
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For a 0, b 1, we denote Wn Un and so Xn Vn , respectively. When p 1, Un Fn
nth Fibonacci number and Vn Ln nth Lucas number.
Inspired by the well-known identity
j
Fn2 Fj Fj1 ,
1.3
n1
Clary and Hemenway 1 obtained factored closed-form expressions for all sums of the form
j
3
n1 Fmn , where m is an integer. Motivated by the results in 1, Melham 2 computed
j
j
4
all sums of the form n1 −1n Fmn
and n1 −1n L4mn . In 3, Melham computed various
n1
nonalternating sums, alternating sums, and sums that alternate according to −1 2 for
sequences {Wn } and {Xn }. The author gathers his sums in three sets. Here we recall one
example from each set for the reader’s convenience:
⎧
1
⎪
⎪
Vj−i1/2 Wji1/2 Wji−1/2
⎪
j
⎨
p
Wn ⎪
1
⎪
ni
⎪
⎩ Uj−i1/2 Xji1/2 Xji−1/2
p
4j3
−1
n1
2
W2n n4i
4j3
−1
n1
2
Un Xn n4i2
if j − i ≡ 1 mod 4,
if j − i ≡ 3 mod 4,
p
U4j−4i4 X4j4i3 ,
Δ−2
1.4
p
V4j−4i5 W4j4i3 2W0 .
Δ−2
We refer to 4 for general expansion formulas for sums of powers of Fibonacci and
Lucas numbers, as considered by Melham, as well as some extensions such that
n
2m
F2kδ
,
n
k0
k0
L2m
2kδ ,
1.5
where δ, ∈ {0, 1}.
For alternating analogues of the results given by Prodinger, that is,
n
k0
2m
,
−1k F2kδ
n
−1k L2m
2kδ ,
1.6
k0
we refer to 5.
Hendel 6 gave the factorization theorem which exhibits factorizations of sums of the
Faj−b . The author also introduced a unified proof method based on formulae for
form ni−1
ji
the factorizations of Fq−d Fqd .
Discrete Dynamics in Nature and Society
3
In 7, Curtin et al. derived formulae for the shifted summations
d−1
Fnj Fmj,
j0
d−1
Lnj Lmj ,
d−1
j0
j0
Fnj Lmj ,
1.7
and the shifted convolutions
d−1
Fnj Fd−m−j ,
d−1
j0
j0
d−1
Lnj Ld−m−j ,
Fnj Ld−m−j
1.8
j0
for positive integers d and arbitrary integers n and m.
In this paper, our main purpose is to consider Melham’s sums involving double
products of terms of {Wn },{Xn },{Un }, and {Vn } given in 3 and then compute several
more general
nonalternating sums, alternating sums, and sums that alternate according to
−1
n1
2
.
2. Certain Finite Sums of Double Products of Terms
In this section, we will investigate certain sums consisting of products of at most two terms
of
n1
{Wn }: nonalternating sums, alternating sums and sums that alternate according to −1 2 .
From the Binet forms of {Wn } and {Xn }, we give the following lemma for further use without
proof.
Lemma 2.1. Let a, b, and p be as in Section 1, and let r aW2 − bW1 . Then for all integers k,
b2 U2k 2abU2k−1 a2 U2k−2 Wk Xk ,
b2 U2k1 2abU2k a2 U2k−1 Wk1 Xk −1k r,
2.1
b V2k 2abV2k−1 a V2k−2 2
2
Xk2
k
−1 2r,
b2 V2k1 2abV2k a2 V2k−1 Xk Xk1 −1k pr.
Theorem 2.2. Fix integers c, d, and m.
i If m is even, then for all integers j > i,
j
ni
Umnc Wmnd
Umj−i1 Xmjicd −1c j − i 1 Xd−c
.
−
ΔUm
Δ
2.2
4
Discrete Dynamics in Nature and Society
i If m is odd, then for all integers j > i,
j
Umnc Wmnd
⎧
Umj−i1 Wmjicd
⎪
⎪
⎪
⎨
Vm
⎪
⎪ Vmj−i1 Xmjicd −1cj Xd−c
⎪
⎩
−
ΔVm
Δ
ni
if j − i ≡ 1 mod 2,
2.3
if j − i ≡ 0 mod2.
Proof. Using the Binet formulas, we compute
j
Umnc Wmnd ni
j mnc
− βmnc
α
α−β
ni
1
α−β
2
j ni
Aαmnd − Bβmnd
α−β
−1mnc d−c
d−c
Aα2mncd Bβ2mncd − 2 Aα Bβ
α−β
2.4
j
j
1
−1c
Xd−c −1mn .
X2mncd −
Δ ni
Δ
ni
Since Xn Wn−1 Wn1 , we can obtain that for even m
j
Umj−i1 Xmjit
X2mnt .
Um
ni
2.5
The result follows.
For example, when i 2, m 3, a 0, b c p 1, and d 5, we obtain
j
F3n1 F3n5 n2
Theorem 2.3. Fix integers c, d, and m. Let S j
F3j−1 F3j4
.
4
ni −1
n
2.6
Umnc Wmnd .
1 If m is odd, then S equals
S
−1j Umj−i1 Xmjicd −1c j − i 1 Xd−c
.
−
ΔUm
Δ
2.7
2 If m is odd and the parities of i and j are the same, then S equals
−1j −1i −1c Xd−c
−1j Vmj−i1 Xmjicd
.
−
ΔVm
2Δ
2.8
Discrete Dynamics in Nature and Society
5
3 If m is odd and the parities of i and j are the different, then S equals
−1j −1i −1c Xd−c
−1j Umj−i1 Wmjicd
.
−
Vm
2
2.9
Proof. Consider
j
n
−1 Umnc Wmnd j mnc
Aαmnd − Bβmnd
− βmnc
α
ni
α−β
ni
α−β
j
−1mnc n
2mncd
d−c
d−c
Bβ2mncd − 2 −1 Aα
2 Aα Bβ
α − β ni
α−β
1
j
j
1
1
−1n X2mncd − Xd−c −1m1nc .
Δ ni
Δ
ni
2.10
Since Xn Wn−1 Wn1 , for odd m, we find
j
−1j Umj−i1 Xmijc
.
−1n X2mnc Um
ni
The result is now obtained by considering the values of
j
ni
2.11
−1m1nc .
Theorem 2.4. Fix integers c, d, and m. For all integers j > i,
4j
−1
n1
2
Umnc Wmnd
n4i1
4j3
−1
n1
2
⎧
⎪V W
U4mj−i ⎨ m sm
V2m ⎪
⎩U X
m sm
Umnc Wmnd
n4i
⎧
⎪U X
U4mj−i1 ⎨ m s3m
V2m ⎪
⎩V W
m s3m
if m is even,
if m is odd,
if m is even,
2.12
if m is odd,
6
Discrete Dynamics in Nature and Society
4j
−1
n4i3
−1
n4i2
Umnc Wmnd
⎧
Vm V2m2j−i−1 Xs3m 2−1c Xd−c
⎪
⎪
⎪
−
⎨
ΔV2m
Δ
⎪
U
V
W
⎪
m 2m2j−i−1 s3m
⎪
⎩
V2m
4j3
n1
2
n1
2
if m is even,
if m is odd,
Umnc Wmnd
⎧
Um V2m2j−i1 Ws5m
⎪
⎪
⎪
⎪
⎪
V2m
⎨
Vm Vm4j−i1−2 Xs5m 2−1c X
d−c
⎪
⎪
⎪
⎪
ΔV
Δ
⎪
2m
⎩
if m is even,
if m is odd,
2.13
where s m4j i c d.
Proof. Consider
4j
−1
n1
2
Umnc Wmnd
n4i1
4j
−1
n1
2
n4i1
1
α−β
2
4j
αmnc − βmnc
α−β
−1
n1
2
Aαmnd − Bβmnd
α−β
Aα2mncd Bβ2mncd
n4i1
−1mnc d−c
d−c
− 2 Aα Bβ
α−β
2.14
4j
4j
n1
n1
1 1
−1 2 X2mncd − Xd−c −1c
−1 2 .
Δ n4i1
Δ
n4i1
Here we have that
4j
n4i1 −1
4j
−1
n4i1
n1
2
n1
2
0 and, by Xn Wn−1 Wn1 ,
X2mncd ΔVm U4mj−i Wm4ji1cd
,
V2m
2.15
for even m. Now formula 2.12 follows. The remaining formulas are proven in a similar
manner.
Notice that in 2.12-2.13, one limit of summation is even while the other is odd.
Accordingly we have observed that each of 2.12-2.13 has a dual sum that is obtained with
Discrete Dynamics in Nature and Society
7
the use of the rule below. We highlight this rule since it also applies to get certain groups of
sums in Section 2.
From 3, we recall the rule for the formation of the dual sum.
1 Replace the even limit by the even limit corresponding to the other residue class
modulo 4 and the odd limit by the odd limit corresponding to the other residue
class modulo 4.
2 Calculate the subscripts on the right in accordance with the paragraph following
2.13.
3 Multiply the right side by −1.
For example, for odd integer m, the dual of 2.13 is
4j1
−1
n1
2
Umnc Wmnd
n4i
1 Vm V2m2j−i1 Xsm
c
−
2−1 Xd−c ,
Δ
V2m
where s is defined as before.
Theorem 2.5. Fix integers c, d, and m.
i If c and d have the same parities, then
4j
−1
n1
2
W2m1nc W2m1nd
n4i1
4j
U2m1 U42m1j−i × X22m1jit X22m1jit1 pr−1t ,
V22m1
−1
n1
2
W2mnc W2mnd
n4i1
4j
V2m U8mj−i × Wm4j4it Xm4j4it ,
V4m
−1
n1
2
W2m1nc W2m1nd
n4i3
U2m1 V22m12j−i−1 × W2m12ji1t1 X2m12ji1t − r−1t ,
V22m1
2.16
8
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4j
−1
n1
2
W2mnc W2mnd
n4i3
2r−1c V
V2m V4m2j−i−1 2
d−c
,
× Xm4ji2t 2r−1t ΔV4m
Δ
4j3
−1
n1
2
W2m1nc W2m1nd
n4i
4j3
V2m1 U42m1j−i1 × W2m12ji1t1 X2m12ji1t − r−1t ,
V22m1
−1
n1
2
W2mnc W2mnd
n4i
4j3
U2m U8mj−i1 2
× Xm4ji2t 2r−1t ,
V4m
−1
n1
2
W2m1nc W2m1nd
n4i2
4j3
2r−1c V
V2m1 V22m12j−i1 d−c
,
× X22m1ji1t1 X22m1ji1t pr−1t −
ΔV22m1
Δ
−1
n1
2
W2mnc W2mnd
n4i2
1 U2m V4m2j−i1 W4mji1t X4mji1t ,
V4m
2.17
where t c d/2 m.
ii If c and d have different parities, then
4j
−1
n1
2
W2m1nc W2m1nd
n4i1
U2m1 U42m1j−i 2
X22m1jiv 2r−1v ,
V22m1
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4j
−1
n1
2
9
W2mnc W2mnd
n4i1
V2m U8mj−i X4mjiv−1 W4mjiv − r−1v ,
V4m
4j
−1
n1
2
W2m1nc W2m1nd
n4i3
1
4j
× U2m1 V22m12j−i−1 W2m12ji1v X2m12ji1v ,
V22m1
−1
n1
2
W2mnc W2mnd
n4i3
V2m V4m2j−i−1 2r−1c Vd−c
,
× Xm4ji2v Xm4ji2v−1 pr−1v ΔV4m
Δ
4j3
−1
n1
2
W2m1nc W2m1nd
n4i
V2m1 U42m1j−i1
× X2m12ji1v W2m12ji1v ,
V22m1
4j3
−1
n1
2
W2mnc W2mnd
n4i
U2m U8mj−i1 × Xm4ji2v Xm4ji2v−1 − pr−1v ,
V4m
4j3
−1
n1
2
W2m1nc W2m1nd
n4i2
−
4j3
2r−1c Vd−c V2m1 V22m12j−i1 2
X22m1ji1v 2r−1v ,
Δ
ΔV22m1
−1
n1
2
W2mnc W2mnd
n4i2
U2m V4m2j−i1 × W4mji1v X4mji1v−1 − r−1v ,
V4m
2.18
where r is defined as before and v c d 1/2 m.
10
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Proof. Suppose that c and d have the same parities. Consider
4j3
−1
n1
2
W2mnc W2mnd
n4i2
4j3
n1 1 A2 α4mncd B2 β4mncd − ABα2mnc β2mnd − ABβ2mnc α2mnd
−1 2
Δ n4i2
1
Δ
b2
4j3
−1
n1
2
V4mncd 2ab
n4i2
a
−1
n1
2
V4mncd−1
n4i2
4j3
2
4j3
−1
n1
2
V4mncd−2
n4i2
4j3
n1
1
c
2
−1 rVd−c
.
−1
Δ
n4i2
2.19
From the definition of {Vn }, we obtain
4j3
−1
n1
2
ΔU2m V4m2j−i1 U2m4ji11c
.
V4m
V4mnc n4i2
Since
4j3
−1
n1
2
2.20
0,
2.21
n4i2
we get
4j3
−1
n1
2
W2mnc W2mnd
n4i2
U2m V4m2j−i1 2
b U2m4ji11cd 2abU2m4ji11cd−1 a2 U2m4ji11cd−2
V4m
2.22
Taking 2k 2m4j i 1 1 c d in Lemma 2.1, we write
4j3
−1
n1
2
W2mnc W2mnd
n4i2
2.23
U2m V4m2j−i1 Wm4ji11dc/2 Xm4ji11dc/2
.
ΔV4m
Thus the result follows. Similar arguments yield the remaining formulas, where we must
consider the parities of c, d.
Discrete Dynamics in Nature and Society
11
For example, the dual of 2.17 is given by if c and d have the same parities,
4j1
−1
n1
2
W2mnc W2mnd −
n4i
U2m V4m2j−i1 W4mjit X4mjit
,
V4m
2.24
and the dual of 2.18 is given by if c and d have different parities,
4j1
−1
n1
2
W2mnc W2mnd
n4i
U2m V4m2j−i1 −
× W4mjiv X4mjiv−1 − r−1v ,
V4m
2.25
where t and v are defined as before.
Acknowledgment
Thanks are due to the anonymous referee who made suggestions towards a better presentation.
References
1 S. Clary and P. D. Hemenway, “On sums of cubes of Fibonacci numbers,” in Applications of Fibonacci
Numbers, vol. 5, pp. 123–136, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
2 R. S. Melham, “Alternating sums of fourth powers of Fibonacci and Lucas numbers,” The Fibonacci
Quarterly, vol. 38, no. 3, pp. 254–259, 2000.
3 R. S. Melham, “Certain classes of finite sums that involve generalized Fibonacci and Lucas numbers,”
The Fibonacci Quarterly, vol. 42, no. 1, pp. 47–54, 2004.
4 H. Prodinger, “On a sum of Melham and its variants,” The Fibonacci Quarterly, vol. 46-47, no. 3, pp.
207–215, 2008.
5 E. Kılıç, N. Ömür, Y. T. Ulutaş et al., “Alternating sums of the powers of Fibonacci and Lucas numbers,”
Miskolc Mathematical Notes, vol. 12, no. 1, pp. 87–103, 2011.
Faj−b ,” The Fibonacci Quarterly, vol. 45, no. 2, pp. 128–132, 2007.
6 R. J. Hendel, “Factorizations of ni−1
ji
7 B. Curtin, E. Salter, and D. Stone, “Some formulae for the Fibonacci numbers,” The Fibonacci Quarterly,
vol. 45, no. 2, pp. 171–179, 2007.
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