INTEGERS 10 (2010), 801-806
#A57
ON BINOMIAL SUMS FOR THE GENERAL SECOND ORDER
LINEAR RECURRENCE
Emrah Kiliç
Mathematics Dept., TOBB Economics and Technology University, Ankara, Turkey
ekilic@etu.edu.tr
Elif Tan
Mathematics Department, Ankara University, Ankara Turkey
etan@ankara.edu.tr
Received: 5/30/10, Accepted: 8/24/10, Published: 12/6/10
Abstract
In this short paper we establish identities involving sums of products of binomial
coefficients and coefficients that satisfy the general second–order linear recurrence.
We obtain generalizations of identities of Carlitz, Prodinger and Haukkanen.
1. Introduction
There are many types of identities involving sums of products of binomial coefficients
and Fibonacci or Lucas numbers. For example, we recall that (see [1, 4, 12]):
n " #
!
n
Fk
k
k=0
n " #
!
n n−k
2
F5k
k
k=0
n " #
!
n
F4k = 3n F2n ,
k
k=0
n " #
!
n n−k
n
= 5 F2n ,
3
F6k = 8n F2n .
k
= F2n ,
(1)
(2)
k=0
Furthermore, many additional sums were given in [2, 8].
As more generalizations of the identities given by (1)-(2), Carlitz [1] derived
the following nice result by ordinary generating functions. If s, t are fixed positive
integers such that s != t, then
n
λ Gsn+r
n " #
!
n k
=
µ Gtk+r
k
(3)
k=0
if and only if
s
λ = (−1)
Ft
s Fs
and µ = (−1)
,
Ft−s
Ft−s
where Gn is either a Fibonacci or Lucas number.
(4)
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INTEGERS: 10 (2010)
Clearly for positive integers s and t, s != t,
Ftn Gsn+r =
n " #
!
n
k=0
k
s(n−k)
(−1)
n−k
Fsk Ft−s
Gtk+r .
(5)
By using the exponential generating functions (or egf’s, see [3, 5, 6, 11]), Prodinger
[11] and Haukkanen [7] obtained the same results as Carlitz [1]. Haukkanen obtained
similar results for the Pell and Pell-Lucas numbers.
The egf of a sequence {an } is defined by
â (x) =
∞
!
n=0
an
xn
.
n!
The product of the egf’s of {an } and {bn } generates the binomial convolution of
{an } and {bn } :
$ n " #
%
∞
!
! n
xn
â (x) b̂ (x) =
an−k bk
.
(6)
k
n!
n=0
k=0
A special case of (5) can be found in [12]. Here the author obtains this special
case by the binomial theorem.
The general recurrence {Wn (a, b; p, q)} is defined, for n ≥ 2, by
Wn = pWn−1 − qWn−2 ,
(7)
where W0 = a, W1 = b.
We write Wn = Wn (a, b; p, q). Let α and β be the roots of λ2 − pλ + q = 0,
assumed distinct. The Binet form of {Wn } is as follows:
Wn = Aαn + Bβ n
(8)
aα−b
where A = b−aβ
α−β and B = α−β .
Define Un = Wn (0, 1; p, q) and Vn = Wn (2, p; p, q). The Binet forms of Un and
Vn are given by
αn − β n
Un =
and Vn = αn + β n
α−β
where {Un } and {Vn } are the generalized Fibonacci and Lucas-types sequences,
respectively.
For more details and properties related to the sequence {Wn }, we refer to [9, 10].
In this short paper, we derive generalizations of the results of [1, 11, 7] for the
sequence {Wn }. Further, some new applications are also given.
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INTEGERS: 10 (2010)
2. The Results for the Sequence {Wn }
We recall the following results from [7]:
Lemma 1. Let λ1 and λ2 be distinct complex numbers, and let c1 and c2 be nonzero
distinct complex numbers. Then
c1 eλ1 x + c2 eλ2 x = c1 eµ1 x + c2 eµ2 x
if and only if
µ1 = λ1 and µ2 = λ2 .
Lemma 2. Let λ1 and λ2 be distinct complex numbers, and let c be a nonzero
complex number. Then
ceλ1 x + ceλ2 x = ceµ1 x + ceµ2 x
if only if either
µ1 = λ1 and µ2 = λ2 or µ1 = λ2 and µ2 = λ1 .
For the sequence {Wn }, we can deduce
Ŵ (x) = Aeαx + Beβx .
Thus we have the following two cases: r != 0 and r = 0.
Theorem 3. Let c and d be nonzero integers and, let r be a nonzero integer. Then
for n ≥ 0
n " #
!
n n−k k
Wcn+r =
t
s Wdk+r
(9)
k
k=0
if and only if
s=
Uc
Ud−c
and t = q c
.
Ud
Ud
Proof. By the egf’s, (9) can be rewritten as
&
'
c
c
d
d
Aαr eα x + Bβ r eβ x = etx Aαr eα sx + Bβ r eβ sx
(10)
where the right-hand side comes from (6). Since αr != β r for r != 0, by Lemma 1,
(10) holds if and only if
αc = αd s + t and β c = β d s + t,
(11)
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INTEGERS: 10 (2010)
and clearly,
s=
αc − β c
Uc
αc − β c
Ud−c
=
and t = αc − αd d
= qc
.
d
d
α −β
Ud
α − βd
Ud
Thus the proof is complete.
Theorem 4. Let c and d be nonzero integers and p = 2b/a. Then for n ≥ 0
Wcn =
n " #
!
n n−k k
t
s Wdk
k
(12)
k=0
if and only if either (11) holds or
s=
−Uc
Ud+c
and t =
.
Ud
Ud
Proof. In terms of the egf’s, (12) could be rewritten as
&
'
c
c
d
d
Aeα x + Beβ x = etx Aeα sx + Beβ sx
(13)
where the right-hand side is seen from (6). Since p = 2b/a, A = B, and thus (13)
takes the form
& d
'
c
c
d
eα x + eβ x = etx eα sx + eβ sx .
(14)
By Lemma 2, (14) holds if and only if either (11) holds or
αc = β d s + t and β c = αd s + t,
so that, clearly,
s=
c
c
αc − β c
−Uc
Ud+c
c
dα −β
=
and
t
=
α
−
β
=
.
d
d
d
d
β −α
Ud
β −α
Ud
Thus, the proof is complete.
From Theorems 3 and 4, we have the following consequence.
Corollary 5. If c and d are nonzero integers and r is an integer, then
n " #
!
n c(n−k) n−k k
Udn Wcn+r =
q
Ud−c Uc Wdk+r .
k
k=0
If c and d are nonzero integers, then
n " #
!
n
k n−k k
Udn Wcn =
(−1) Ud+c
Uc Wdk .
k
k=0
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INTEGERS: 10 (2010)
We note the following known special cases of {Wn } :
p
1
1
2
2
1
1
q
−1
a
0
b
1
Wn
Fn
Fibonacci numbers
−1
2
1
Ln
Lucas numbers
−1
0
1
Pn
Pell numbers
−1
2
2
2
Pell-Lucas numbers
−2
0
1
Jn
Jacobsthal numbers
−2
2
1
jn
Jacobsthal-Lucas numbers
Thus we have the following examples:
Fdn Fcn+r
=
n " #
!
n
k=0
and
Fdn Lcn
=
k
c(n−k)
(−1)
n " #
!
n
k
k=0
n−k k
Fd−c
Fc Fdk+r
k
n−k k
(−1) Fd+c
Fc Ldk ,
which are also given in [1, 11, 7].
Similar to the Fibonacci and Lucas numbers, for the Jacobsthal and JacobsthalLucas sequences, we obtain
Jdn Jcn+r
Jdn jcn
=
=
n " #
!
n
k=0
n "
!
k=0
k
c(n−k)
(−2)
n−k k
Jd−c
Jc Jdk+r ,
#
n
k n−k k
(−1) Jd+c
Jc jdk .
k
References
[1] L. Carlitz, Some classes of Fibonacci sums, Fibonacci Quart. 16 (1978), 411–426.
[2] L. Carlitz and H. H. Ferns, Some Fibonacci and Lucas identities, Fibonacci Quart. 8 (1970),
61–73.
[3] C. A. Church and M. Bicknell, Exponential generating functions for Fibonacci identities,
Fibonacci Quart. 11 (1973), 275–281.
[4] R. A. Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing Co.
River Edge, NJ, 1997.
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806
[5] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for
Computer Science, Reading Mass: Addision-Wesley, 1989.
[6] R. T. Hansen, General identities for linear Fibonacci and Lucas summations, Fibonacci Quart.
16 (1978), 121–128.
[7] P. Haukkanen, On a binomial sum for the Fibonacci and related numbers, Fibonacci Quart.
34 (1996), 326–331.
[8] P. Haukkanen, Formal power series for binomial sums of sequences of numbers, Fibonacci
Quart. 31 (1993), 28–31.
[9] A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci
Quart. 3 (1965), 161-176.
[10] E. Kilic and P. Stanica, Factorizations of binary polynomial recurrences by matrix methods,
Rocky Mount. J. Math, to appear.
[11] H. Prodinger, Some information about the Binomial transform, Fibonacci Quart. 32 (1994),
412-415.
[12] S Vajda, Fibonacci and Lucas numbers, and the Golden Section: Theory and Applications,
John Wiley & Sons, Inc, New York, 1989.