Zhang and Wu Advances in Difference Equations 2013, 2013:377
http://www.advancesindifferenceequations.com/content/2013/1/377
RESEARCH
Open Access
On the reciprocal sums of the generalized
Fibonacci sequences
Han Zhang and Zhengang Wu*
*
Correspondence:
sky.wzgff[email protected]
Department of Mathematics,
Northwest University, Xi’an, Shaanxi,
P.R. China
Abstract
The Fibonacci sequence has been generalized in many ways. One of them is defined
by the relation un = aun–1 + un–2 if n is even, un = bun–1 + un–2 if n is odd, with initial
values u0 = 0 and u1 = 1, where a and b are positive integers. In this paper, we
consider
the reciprocal sum of un and then establish some identities relating to
1 –1
( ∞
k=n uk ) , where x denotes the nearest integer to x.
MSC: Primary 11B39
Keywords: generalized Fibonacci sequence; infinite sum; reciprocal sum
1 Introduction
For any integer n ≥ , the well-known Fibonacci sequence Fn is defined by the secondorder linear recurrence sequence Fn+ = Fn+ + Fn , where F =  and F = . The Fibonacci
sequence has been generalized in many ways, for example, by changing the initial values,
by changing the recurrence relation, and so on. Edson and Yayenie [] defined a further
generalized Fibonacci sequence un depending on two real parameters used in a non-linear
recurrence relation, namely,
aun– + un–
un =
bun– + un–
if n is even and n ≥ ,
if n is odd and n ≥ ,
()
with initial values u =  and u = , where a, b are positive integers. This new sequence
is actually a family of sequences where each new choice of a and b produces a distinct
sequence. When a = b = , we have the classical Fibonacci sequence and when a = b = ,
we obtain the Pell numbers. Even further, if we set a = b = k for some positive integer k,
we obtain the k-Fibonacci numbers.
Various properties of the Fibonacci numbers and related sequences have been studied by
many authors, see [–]. Recently, Ohtsuka and Nakamura [] studied the partial infinite
sums of reciprocal Fibonacci numbers and proved that
∞
– 
if n is even and n ≥ ,
Fn–
=
Fk
F
–

if
n is odd and n ≥ ,
n–
k=n
where x (the floor function) denotes the greatest integer less than or equal to x.
Some related works can also be found in [–]. In particular, in [], the authors studied a problem which is a little different from that of [], namely that of determining the
©2013 Zhang and Wu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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Zhang and Wu Advances in Difference Equations 2013, 2013:377
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
 –
nearest integer to ( ∞
k=n vk ) . Specifically, suppose that x = x +  (the nearest integer
function), and {vn }n≥ is an integer sequence satisfying the recurrence formula
vn = a vn– + a vn– + · · · + as vn–s
(s ≥ )
for any positive integer a , a , . . . , am , with the initial conditions v ≥ , vk ∈ N,  ≤ k ≤
s – . Then, provided a ≥ a ≥ · · · ≥ am ≥ , we can conclude that there exists a positive
integer n such that
∞
 – = vn – vn–
vk
k=n
for all n > n .
Because the Fibonacci sequence has been generalized to a higher-order recursive sequence, any study on linear recursive sequences has little significance in this context, and
we have to consider other non-linear recursive sequences. The main purpose of this paper
is concerned with finding expressions for
∞
 – .
uk
k=n
In fact, this problem is difficult because each item of this sequence relies on the previous
relation. In order to resolve the question, we consider the reciprocal sums in two directions: on the one hand to the subsequence upk+q and on the other to the product form
uk uk+c+ , where p, q, c are non-negative integers and p ≥ . The results are as follows.
Theorem  Let {un } be a second-order sequence defined by (). For any even p ≥  and
non-negative integer q < p, there exists a positive integer n such that
∞
– 
= upn+q – upn–p+q
upk+q
k=n
for all n ≥ n .
Theorem  Let {un } be a second-order sequence defined by (). For any integer c ≥ , there
exists a positive integer n such that
∞
ak bk+c+ – u u
un– un+c n n+c+
–
–
=
uk uk+c+
an bn+c+ an– bn+c k=n
for all n ≥ n .
Open problem In the light of our investigation, for any positive integer s ≥  and l,
whether there exist identities for
∞

usk
k=n
–
and
∞
k=n

uk uk+l
–
represent two interesting, albeit challenging, open problems.
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Zhang and Wu Advances in Difference Equations 2013, 2013:377
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2 Proofs of the theorems
We need the following lemma.
Lemma (Generalized Binet’s formula) The terms of the generalized Fibonacci sequence un
are given by
n
un =
a  –n+ α n – β n
,
·
n
α–β
(ab)  where α =
√
ab+ a b +ab
,

√
ab– a b +ab
.

β=
Proof See Theorem  of [].
Proof of Theorem  From the geometric series as → , we have

=  ∓ + O  =  + O().
±
From Lemma and the identity αβ = –ab, we have
⎧
⎪
⎨
upk+q =
⎪
⎩
α pk+q –β pk+q
if q is even (so that pk + q is even),
pk+q– pk+q
a  b  (α–β)
α pk+q –β pk+q
ab
if q is odd (so that pk + q is odd).
pk+q–

(α–β)
Let
⎧
⎪
⎨
A=
⎪
⎩

a
if q is even,
pk+q– pk+q

b  (α–β)

ab
if q is odd.
pk+q–

(α–β)
Thus,
upk+q = Aα pk+q + O
pk
|β| 
α
.
pk

Hence,

upk+q

=
=
pk

Aα pk+q ( + O( |β|pk ))
pk |β| 


+
O
pk
pk+q
Aα
α 
α 
=
pk
|β| 

+O
pk
pk+q
Aα
α 
.
Thus,
∞
k=n

upk+q
=

α pn+q
A( –

)
αp
+O
pn
|β| 
α
pn

=
pn |β| 
αp

+
O
.
pn
pn+q
p
Aα
(α – )
α 
Zhang and Wu Advances in Difference Equations 2013, 2013:377
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Taking the reciprocal of this expression yields
∞
–

upk+q
k=n
pn Aα pn+q (α p – )
|β| 
+
O
pn
p
α
α
pn |β| 
= Aα pn+q – Aα pn–p+q + O
pn
α
pn |β| 
= upn+q – upn–p+q + O
.
pn
α
=
()
Therefore, for any even p ≥  and integer  < q < p, there exists n ≥ n sufficiently large
such that the modulus of the last error term of identity () becomes less than /. This
completes the proof of Theorem .
Proof of Theorem  In the first place, suppose that k ≥  is even. From Lemma we have
uk =

a
k –

b
k

·
αk – β k
,
α–β
and
uk+c+ =

k
k
a  +c b  +c
·
α k+c+ – β k+c+
.
α–β
The identities (α – β) = a b + ab and αβ = –ab now yield
α k+c+ + β k+c+ – (αβ)k (α c+ + β c+ )
ak+c– bk+c (α – β)
k α k+c+
αβ
=
+
O

k+c
(ab + b)(ab)
ab
uk uk+c+ =
=
α k+c+
+ O().
(ab + b)(ab)k+c
Further, if k ≥  is odd, the same identity is similarly obtained. Thus, in both cases we have

uk uk+c+
=
=

α k+c+
(ab +b)(ab)k+c
+ O()
=

k
α k+c+
( + O( (ab)
))
(ab +b)(ab)k+c
α k
(ab)k
(ab + b)(ab)k+c
+
O
.
α k+c+
α k
Hence,
(ab)k
ak bk+c+ (ab + b )ak+c bk+c
=
+O
uk uk+c+
α k+c+
α k
ac+ bc+ + ac bc+ (ab)k
(ab)k
=
· k + O
.
α c+
α
α k
Zhang and Wu Advances in Difference Equations 2013, 2013:377
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ac+ bc+ +ac bc+
,
α c+
Let B =
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then
(ab)k
ak bk+c+ B(ab)k
=
+O
.
uk uk+c+
α k
α k
Consequently,
∞
ak bk+c+
k=n
uk uk+c+
=B
∞
(ab)k
α k
k=n
+O
∞
(ab)k
α k
k=n
(ab)n
=B·
+O
.
α n
 – ( ab
)
α
)n
( ab
α
Taking the reciprocal of this expression yields
∞
ak bk+c+
k=n
–
=
uk uk+c+
=

B·
n
( ab
α )

–( ab
α )
)
 – ( ab
α
B · ( ab
)n
α
( + O( |β|
))
αn
n
+O
|β|n
αn

+O n n
=
α |β|
B · ( ab
)n
α
n
n–
α
α



– ·
+O n n .
= ·
B
ab
B
ab
α |β|
)
 – ( ab
α
()
On the other hand,
un un+c+ un– un+c
–
an bn+c+ an– bn+c
n
n–
α c+
α
α c+
α

·
–
·
+
O
ac+ bc+ + ac bc+
ab
ac+ bc+ + ac bc+
ab
a n bn
n
n–
α
α



= ·
– ·
+O n n .
()
B
ab
B
ab
a b
=
Combining () and (), finally we have
∞
ak bk+c+
k=n
uk uk+c+
–
un un+c+ un– un+c
–
–
an bn+c+ an– bn+c

=O n n .
a b
()
It follows that for any integer c ≥ , there exists n ≥ n sufficiently large such that the
modulus of the last error term of identity () becomes less than /. This completes the
proof of Theorem .
Competing interests
The authors declare that they have no competing interests.
Zhang and Wu Advances in Difference Equations 2013, 2013:377
http://www.advancesindifferenceequations.com/content/2013/1/377
Authors’ contributions
HZ obtained the theorems and completed the proof. ZW corrected and improved the final version. Both authors read
and approved the final manuscript.
Acknowledgements
The authors express their gratitude to the referee for very helpful and detailed comments. This work is supported by the
N.S.F. (11371291), the S.R.F.D.P. (20136101110014), the N.S.F. (2013JZ001) of Shaanxi Province, and the G.I.C.F. (YZZ12062) of
NWU, P.R. China.
Received: 16 October 2013 Accepted: 1 December 2013 Published: 23 Dec 2013
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10.1186/1687-1847-2013-377
Cite this article as: Zhang and Wu: On the reciprocal sums of the generalized Fibonacci sequences. Advances in
Difference Equations 2013, 2013:377
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