Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 684280, 7 pages
doi:10.1155/2012/684280
Research Article
On Reciprocal Series of Generalized Fibonacci
Numbers with Subscripts in Arithmetic Progression
Neşe Ömür
Department of Mathematics, Kocaeli University, 41380 Izmit, Turkey
Correspondence should be addressed to Neşe Ömür, neseomur@kocaeli.edu.tr
Received 24 October 2011; Accepted 16 November 2011
Academic Editor: Elena Braverman
Copyright q 2012 Neşe Ömür. 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.
∞
We investigate formulas for closely related series of the forms:
n0 1/Uanb c,
∞
2
n
2
∞
2
−1
U
/U
c
,
U
/U
c
for
certain
values
of
a,
b, and c.
anb
anb
2anb
n0
n0
anb
1. Introduction
Let p be a nonzero integer such that Δ p2 4 /
0. The generalized Fibonacci and Lucas
sequences are defined by the following recurrences:
Un1 pUn Un−1 ,
Vn1 pVn Vn−1 ,
1.1
where U0 0, U1 1 and V0 2, V1 p, respectively. When p 1, Un Fn nth Fibonacci
number and Vn Ln nth Lucas number.
If α and β are the roots of equation x2 − px − 1 0, the Binet formulas of the sequences
{Un } and {Vn } have the forms:
Un αn − βn
,
α−β
Vn αn βn ,
1.2
respectively.
In 1, Backstrom developed formulas for closely related series of the form:
∞
1
n0
Fanb c
,
1.3
2
Discrete Dynamics in Nature and Society
for certain values of a, b, and c. For example, he obtained the following series:
√
∞
1
K 5
,
F
FK
2LK
n0 2n1
⎧ √
⎪
5
−
5F
/L
⎪
t
t
⎪
⎪
∞
⎨
, t even,
1
√ 2LK F
FK ⎪
⎪
n0 2n1K2t
⎪
5 − Lt /Ft
⎪
⎩
, t odd,
2LK
1.4
where K represents an odd integer and t is an integer in the range −K − 1/2 to K − 1/2
inclusive. Also, he gave the similar results for Lucas numbers.
In 2, Popov found in explicit form series of the form:
∞
1
n0
Fanb ± c
∞
,
n0
∞
1
,
Fanb Fcnd
2
n0 Fanb
1
,
2
± Fcnd
1.5
for certain values of a, b, c, and d.
In 3, Popov generalized some formulas of Backstrom 1 related to sums of reciprocal
series of Fibonacci and Lucas numbers. For example,
⎧ s
β
β
⎪
⎪
< 1,
,
⎪
snr
∞
⎨
−q
Ur Us α 1.6
Δ
snr
α
αs
Vr ⎪
⎪
n0 V2n1r2s − −q
⎪
, < 1,
⎩
Ur Us
β
where s and r are integers.
In 4, Gauthier found the closed form expressions for the following sums:
m
−1kn f2k1n
k1
2
2
fk1n
fkn
m
−1kn f2k1n
k0
2
2
lk1n
lkn
m, n ≥ 1,
,
1.7
m, n ≥ 0,
,
where for x / 0 an indeterminate, the generalized Fibonacci and Lucas polynomials {fn }n and
{ln }n are given by the following recurrences:
fn2 xfn1 fn ,
ln2 xln1 ln ,
f0 0, f1 1, n ≥ 0,
1.8
l0 2, l1 x, n ≥ 0,
respectively.
In this paper, we investigate formulas for closely related series of the forms:
∞
1
n0
Uanb c
,
for certain values of a, b and c.
∞
−1n Uanb
n0
2
Uanb c
,
∞
n0
U2anb
2
c
Uanb
2 ,
1.9
Discrete Dynamics in Nature and Society
3
2. On Some Series of Reciprocals of Generalized Fibonacci Numbers
In this section, firstly, we will give the following lemmas for further use.
Lemma 2.1. Let n be an arbitrary nonzero integer. For integer m ≥ 1,
m
−1kn U2k1n
k1
2
2
Uk1n
Ukn
1
4Un
Vn2
Un2
−
2
Vm1n
2
Um1n
,
2.1
and for integer m ≥ 0,
m
−1kn U2k1n
2
2
Vk1n
Vkn
k0
2
Um1n
2
4Un Vm1n
.
2.2
Proof. We give the proof of Lemma 2.1 as the proofs of the sums in 4, using the following
equalities:
Vk1n
U2k1n
1 Vkn
,
Uk1n Ukn 2 Ukn Uk1n
U2k1n
1 Uk1n Ukn
,
Vk1n Vkn 2 Vk1n
Vkn
2.3
Vk1n
1 Vkn
−1kn Un
−
,
Uk1n Ukn 2 Ukn Uk1n
1 Uk1n Ukn
−1kn Un
−
.
Vk1n Vkn 2 Vk1n
Vkn
Lemma 2.2. For arbitrary integers n and t,
V2n − −1n−t V2t ΔUn−t Unt ,
V2n −1n−t V2t Vn−t Vnt ,
Un2 − −1n−t Ut2 Un−t Unt ,
Vn2 − −1n−t Vt2 ΔUn−t Unt .
2.4
2.5
Proof. From Binet formulas of sequences {Un } and {Vn }, the desired results are obtained.
Theorem 2.3. For an odd integer t,
m
1
n1
U2n1t Ut
m
1
n1
1
2Vt
1
U2n1t − Ut 2Vt
2 − V2t 2 − V2m1t
−
,
U2t
U2m1t
2 V2t 2 V2m1t
−
.
U2t
U2m1t
2.6
4
Discrete Dynamics in Nature and Society
Proof. By replacing n with 2n 1t in 2.5, we have
2
− Ut2 U2nt U2n1t ,
U2n1t
or
1
U2n1t Ut
2.7
U2n1t − Ut
.
U2nt U2n1t
2.8
Taking r 2n 1t and s t in the equality Vs Ur Urs −1s Ur−s 5, the equality 2.8 is
rewritten as follows:
1
Ut
1
1
−1t
−
.
2.9
U2n1t Ut Vt U2nt U2n1t
U2nt U2n1t
We have the sum
m
n1
m
1 U2n1t Ut Vt n1
1
1
−1t
U2nt U2n1t
− Ut
m
n1
1
.
U2nt U2n1t
2.10
For an odd integer t, we have
m 1
1
1
1
−
−
,
U2nt U2n1t
U2t U2m1t
n1
2.11
and taking s 2nt and r 2t in identity 5:
Usr Vs − Us Vsr 2−1s Ur ,
we get
m
n1
m V2n1t
V2m1t
1
1 V2nt
1
V2t
−
−
.
U2nt U2n1t 2U2t n1 U2nt U2n1t
2U2t U2t U2m1t
2.12
2.13
Substituting 2.11 and 2.13 in 2.10, we have the desired result.
For example, if we take t 1 and p 1 in 2.6, we have
m
n1
1
F2m1 − 1
.
F2n1 1
F2m1
2.14
Note that
F2m1
m
1
n1
F2n1 1
m
n1
F2n .
2.15
Discrete Dynamics in Nature and Society
5
Corollary 2.4. For an odd integer t,
⎧
Vm1t
Vt
1
⎪
⎪
−
,
m is
⎪
⎪
m
⎨ 2Vt Um1t Ut
1
U2n1t Ut ⎪
⎪
1 ΔUm1t Vt
n1
⎪
⎪
−
, m is
⎩
2Vt
Vm1t
Ut
⎧
Δ Ut Um1t
⎪
⎪
−
,
m is
⎪
⎪
⎨ 2Vt Vt
Vm1t
m
1
⎪
n1 U2n1t − Ut
⎪
1 ΔUt Vm1t
⎪
⎪
−
, m is
⎩
2Vt
Vt
Um1t
even,
odd,
2.16
even,
odd.
Proof. Using the equalities V2n Vn2 − 2−1n ΔUn2 2−1n and U2n Un Vn in Theorem 2.3,
the results are obtained.
Corollary 2.5. Let t be an odd integer. For |β/α| < 1, t > 0 and |α/β| < 1, t < 0,
∞
1
n1
U2n1t Ut
∞
n1
1 √
Vt
Δ−
,
2Vt
Ut
Δ
U2n1t − Ut 2Vt
1
2.17
1
Ut
−√
,
Vt
Δ
and for |β/α| < 1, t < 0 and |α/β| < 1, t > 0,
∞
1
n1
U2n1t Ut
∞
n1
−1 √
Vt
,
Δ
2Vt
Ut
Δ
U2n1t − Ut 2Vt
1
2.18
1
Ut
√
.
Vt
Δ
Proof. Since
lim
n→∞
Vanb
Uanb
⎧
√
⎪
⎪
⎪ Δ
⎪
⎨
⎪
√
⎪
⎪
⎪
⎩− Δ
β
< 1, a > 0,
α
β
< 1, a < 0,
α
α
< 1, a < 0,
β
α
< 1, a > 0,
β
2.19
the results are easily seen by equalities 2.16.
Theorem 2.6. For an integer m ≥ 1 and an arbitrary nonzero integer t,
m
n1
−1nt U2n1t
V2n1t − −1nt Vt
1
2 4Δ2 Ut
Vt2
Ut2
−
2
Vm1t
2
Um1t
.
2.20
6
Discrete Dynamics in Nature and Society
Proof. By replacing n with 2n 1t/2 and t with t/2 in 2.4, we have
V2n1t − −1nt Vt ΔUnt Un1t ,
or
1
nt
V2n1t − −1 Vt
2.21
1
.
ΔUnt Un1t
2.22
Multiplying equality 2.22 by −1nt U2n1t /Unt Un1t , we get
−1nt U2n1t
−1nt U2n1t
.
2
2
ΔUnt
Un1t
Unt Un1t V2n1t − −1nt Vt
2.23
We have the sum:
m
n1
m
−1nt U2n1t
−1nt U2n1t
1 .
2
2
Δ n1 Utn
Utn1
Unt Un1t V2n1t − −1nt Vt
2.24
Using the equalities 2.1 and 2.21, the proof is obtained.
Corollary 2.7. For an arbitrary nonzero integer t,
∞
n1
−1nt U2n1t
V2n1t − −1nt Vt
1
2 4Δ2 Ut
Vt2
Ut2
−Δ .
2.25
Proof. Taking m → ∞ in Theorem 2.6 and using 2.19, the result is easily obtained.
Theorem 2.8. For an integer m ≥ 0 and an arbitrary nonzero integer t,
m
n0
−1nt U2n1t
V2n1t −1nt Vt
2 2
Um1t
2
4Ut Vm1t
.
2.26
Proof. The proof of the theorem is similar to the proof of Theorem 2.6.
Corollary 2.9. For an arbitrary nonzero integer t,
∞
n0
−1nt U2n1t
nt
V2n1t −1 Vt
1
2 4ΔU .
t
2.27
Proof. Taking m → ∞ in Theorem 2.8 and using 2.19, the result is easily obtained.
For example, if we take t 3 and p 1 in 2.27, we have
∞
n0
−1n F32n1
1
2 40 .
L32n1 −1 4
n
2.28
Discrete Dynamics in Nature and Society
7
Theorem 2.10. For an integer m ≥ 1 and an arbitrary nonzero integer t,
m
n1
m
n1
U22n1t
1
2 4U
2t
2
2
U2n1t
− Ut
U22n1t
1
2 2U
4Δ
2t
2
V2t2
−
2
U2t
2
V2n1t
− Vt
V2t2
2
V2m1t
2
U2m1t
2
V2m1t
−
2
U2t
2
U2m1t
,
2.29
.
Proof. The proof of theorem is similar to the proof of Theorem 2.6.
Corollary 2.11. For an arbitrary nonzero integer t,
∞
n1
∞
n1
U22n1t
1
2 4U
2t
2
1
2 2U
4Δ
2t
2
2
V2n1t
− Vt
−Δ ,
2
U2t
2
U2n1t
− Ut
U22n1t
V2t2
V2t2
2
U2t
2.30
−Δ .
Proof. Taking m → ∞ in Theorem 2.10 and using 2.19, the result is easily obtained.
For example, if we take t 2 in the equality 2.30, we have
∞
n1
U42n1
2
V22n1
−
V22
2 1
Δ2 U43
.
2.31
References
1 R. P. Backstrom, “On reciprocal series related to Fibonacci numbers with subscripts in arithmetic progression,” The Fibonacci Quarterly, vol. 19, pp. 14–21, 1981.
2 B. S. Popov, “On certain series of reciprocals of Fibonacci numbers,” The Fibonacci Quarterly, vol. 22, no.
3, pp. 261–265, 1984.
3 B. S. Popov, “Summation of reciprocal series of numerical functions of second order,” The Fibonacci
Quarterly, vol. 24, no. 1, pp. 17–21, 1986.
4 N. Gauthier, “Solution to problem H-680,” The Fibonacci Quarterly, vol. 49, no. 1, pp. 90–92, 2011.
5 S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications, John Wiley & Sons,
New York, NY, USA, 1989.
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