Download Modified k-Pell Sequence: Some Identities and Ordinary

Applied Mathematical Sciences, Vol. 7, 2013, no. 121, 6031 - 6037
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2013.39513
Modified k-Pell Sequence: Some Identities and
Ordinary Generating Function
Paula Catarino1 and Paulo Vasco2
Department of Mathematics
School of Science and Technology
University of Trás-os-Montes e Alto Douro (Vila Real – Portugal)
[email protected], [email protected]
Copyright © 2013 Paula Catarino and Paulo Vasco. 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.
Abstract
Some identities of the k-Pell-Lucas sequence and their relationship to the
Modified k-Pell sequence allow us to obtain some identities for Modified k-Pell
numbers. Also the ordinary generating function for the Modified k-Pell sequence
and another expression for the general term of the sequence, using the ordinary
generating function, is provided.
Mathematics Subject Classification 2010: 11B37, 05A15, 11B83
Keywords: Pell numbers, k-Pell numbers, Modified Pell numbers, Modified k-Pell
numbers, Binet’s formula, Generating Functions
1.
Introduction
In this paper we consider one of the sequences of positive integer’s numbers
satisfying a recursive recurrence and we give some well-known identities for this
type of sequences. One of the sequences of positive integers (also defined
recursively) that have been studied over several years is the well-known Fibonacci
1
Member of CIDMA and Collaborator of CM-UTAD (Portuguese research centers)
2
Member of CM-UTAD
6032
Paula Catarino and Paulo Vasco
(and Lucas) sequence. Many papers are dedicated to Fibonacci sequence, such as
the work of Hoggatt, in [16] and Vorobiov, in [11], among others and more
recently we have, for example, the works of Caldwell et al. in [4], Marques in [7],
Shattuck in [10] and Falcın et al., in [15]. In [15] the authors consider some
properties for the k-Fibonacci numbers obtained from elementary matrix algebra
and its identities including generating function and divisibility properties appears
in the paper of Bolat et al., in [3]. Sometimes, in the literature (see, for example,
Dasdemir in [1]), are considered other sequences namely, Pell, k-Pell, Pell-Lucas,
k-Pell Lucas and Modified Pell sequences, where k is a positive real number. The
sequence of Pell numbers ∈ is defined by the recursive sequence given by
2
, 2, with the initial conditions 0 and 1.
This sequence has been studied and some of its basic properties are known (see,
for example, the study of Horadam, in [2]). For any positive real number k, the
k-Pell sequence , ∈ is defined recurrently (in the similar way as in the
previous sequence), by , 2, + k,
, for 1, with the initial
conditions given by , 0 , , 1 (see [12] and [13]). The Pell-Lucas
sequence ∈ is defined by 2
, 2, with the initial
conditions 2 and the k-Pell-Lucas , ∈ is the sequence
satisfying the recursive sequence given by , 2, + k,
, for 1,
with the initial conditions given by , 2, , 2 (see [14]). The Modified
Pell sequence ∈ is given by the following recursive relation 2
, 2 and the initial conditions are 1, 1.
The Binet’s formula is also well known for several of these sequences. Claude
Levesque, in [5] finds the general Binet’s formula for a general mth order linear
recurrence. Sometimes, some basic properties follows from this formula. For
example, for the sequence of Jacobsthal numbers, Koken and Bozkurt, in [8],
deduce some properties and the Binet’s formula, using the matrix method. In [9],
Yilmaz et al. study some more properties related with k-Jacobsthal numbers.
According Catarino, in [12], Catarino et al., in [14], and also Jhala et al., in [6],
we consider, in this paper, the Modified k-Pell sequence and many properties are
proved by easy arguments for the Modified k-Pell numbers.
2.
The Modified k-Pell sequence and some identities
For any positive real number k, we define the Modified k-Pell , ∈
sequence satisfying the recursive sequence given by
, 2, + k,
, for 1,
(1)
with the initial conditions given by , 1, , 1.
We can find the explicit formula for the term of order n of the Modified k-Pell
sequence using the well-known results involving recursive sequences. Normally
we use the characteristic equation associated to the recurrence relation (1) in order
to obtain the well-known Binet’s formula for this type of sequences. After, as a
consequence we get some properties for the sequence. In this case, the
characteristic equation associated to (1) is the equation
Modified k-Pell sequence
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2 0
(2)
and it is easy to get the Binet’s formula using its two distinct roots. However, since
Claude Levesque, in [5], finds the general Binet’s formula for a general mth order
linear recurrence, we use, for this particular case, the section 2 of [5]. Hence we get,
Proposition 1 (Binet’s formula)
The nth Modified k-Pell number is given by
, ,
where 1 √1 and 1 √1 characteristic equation (2) and > .
(3)
are
the
Now using the Binet’s formula of k-Pell-Lucas , ∈
Proposition 1, in [14]), we immediately conclude that
Proposition 2
roots
of
the
∎
sequence (see,
2, , .
Now we use (4) and obtain other identity satisfied by the sequence , ∈ .
(4)
∎
Proposition 3 (Catalan’s identity)
,
, ,
!"
#,
!" $.
(5)
Proof: Using (4) and the Catalan’s identity of k-Pell-Lucas , ∈ sequence
(see, Proposition 2, in [18]) we get
,
, ,
% #,
, ,
$
% ! "
#,
4!" $
that is, the identity required.
! "
#4,
4!" $
%
!"
#,
! " $,
∎
We observe that using directly the Binet’s formula (3) we also obtain (5) doing
some calculations and proper simplifications.
Note that for 1 in Catalan’s identity obtained, we get the Cassini’s identity for
the Modified k-Pell sequence. In fact, the equation (5) for 1, yields
,
, ,
!"
',
!"(
and using one of the initial conditions of this sequence we obtain the following
result.
Paula Catarino and Paulo Vasco
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Proposition 4 (Cassini’s identity)
,
, ,
! "
!1 ".
(6)
∎
Once more, it is easy to get (6) using directly the Binet’s formula (3). The next
identity (d’Ocagne’s identity) can also be obtained using the Binet’s formula (3) as it
was done in [6] for the k-Jacobsthal sequence, in [12] for the -Pell sequence and
in [14] for the -Pell-Lucas sequence. However, we shall use the Proposition 4 of
[14] and Proposition 2. Hence we have
Proposition 5 (d’Ocagne’s identity)
If m > n then
)
,) , ,) , !1" √1 ',)
#1 √1 $
(.
Proof: Using Proposition 4 of [14] and Proposition 2, we get
,) , ,) , #,) , ,) , $
%
% !1" 2√1 ',)
2#1 √1 $
% !1" 4√1 ',)
#1 √1 $
)
)
)
and the result follows.
!1" √1 ',)
#1 √1 $
(
(
(
∎
Again using the Binet’s formula (3) we can obtain other property of the Modified
k-Pell sequence which is stated in the following proposition. However, we use the
Proposition 2 and the Proposition 5 of [14].
Proposition 6
Proof: We have that
/0,
lim→. /
0,1
/0,
lim→. /
0,1
.
(7)
20,
lim→. 2
0,1
.
∎
Also, we easily can show the following result using basic tools of calculus of limits
and (7).
Proposition 7
/
lim→. /0,1 .
(8)
0,
∎
Modified k-Pell sequence
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3. Generating functions for the Modified k-Pell sequences
Next we shall give the generating functions for the Modified k-Pell sequences.
In this section we use the generating function for the k-Pell Lucas sequence that
was introduced by Catarino et al. in [14]. In this paper, the authors showed that
the ordinary generating function of the k-Pell Lucas sequence is written as
7
3 !4 " ∑.
(9)
6 , 4 7
7 .
Of course, the Modified k-Pell sequence can also be considered as the
coefficients of the power series of the corresponding generating function, as usual
for this type of sequences. Let us suppose that the Modified k-Pell numbers of
order k are the coefficients of a power series centered at the origin, and let us
consider the corresponding analytic function 8 defined by
8 !4" , , 4 , 4 ⋯ , 4 ⋯,
(10)
and called the generating function of the Modified k-Pell numbers.
Using (10) and Proposition 2, we get,
28 !4 " 2, 2, 4 2, 4 ⋯ 2, 4 ⋯
, , 4 , 4 ⋯ , 4 ⋯
Now from (9) we can write (10) as follows
7
8 !4" '
7
7 (.
Therefore, the ordinary generating function of the Modified k-Pell sequence can be
written as
7
8 !4" 7
7 .
(11)
According with Catarino et al. in [14] and Catarino in [12], recall that if for some
;
sequence !: " , it is known that lim→. ;< =, being = a positive real number,
then considering the power series ∑.
6 : 4 , its radius of convergence > is equal
to ?. So, for the Modified k-Pell sequence, using (7) we know that it can be written
as a power series with radius of convergence equal to
. Next we give another
expression for the general term of the Modified k-Pell sequence using the ordinary
generating function.
Proposition 8
Let us consider8 !4" ∑.
6 , 4 , for 4 ∈ @ , A. Then we have that
!"
, B !" !"
!
,
where 8 denotes the n order derivative of the function 8 .
Proof: We have that , 4 1 and
th
8 D !4" ∑.
1 ∑.
.
6 , 4
6 , 4
Also,
Paula Catarino and Paulo Vasco
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8 !" !4" ∑.
2 ∙ 1 ∙ , ∑.
;
6 ! 1", 4
6F ! 1", 4
⋮⋮
⋮
!H"
.
8 !4 " ∑6H ! 1" ⋯ ! !I 1"", 4 H .
So,
8 !H" !4 " I !I 1" ⋯ #I !I 1"$,H ∑.
6H ! 1" ⋯ ! !I !H"
H
1"", 4
and then we get 8 !4" I! ,H ∑.
6H ! 1" ⋯ ! !I 1"", 4 H . Therefore, 8 !H" !0" I! ,H and ,H 1, we have that , B0
B0 !J" !"
H!
!" !"
!
, as required.
Note that using Proposition 7 of [14], we can also write ,
. Hence, for all
∎
in terms of the nth
L0 !" !"
order derivative of the generating function (9), getting , K
!
M.
References
[1] A. Dasdemir, “On the Pell, Pell-Lucas and Modified Pell Numbers by Matrix
Method”, Applied Mathematical Sciences, 5(64) (2011), 3173-3181.
[2] A. F. Horadam, “Pell identities”, The Fibonacci Quarterly, 9(3) (1971)
245-252, 263.
[3] C. Bolat, H. Köse, “On the Properties of k-Fibonacci Numbers”, Int. J.
Contemp. Math. Sciences, 5(22) (2010), 1097-1105.
[4] C. K. Caldwell, T. Komatsu, “Some Periodicities in the Continued Fraction
Expansions of Fibonacci and Lucas Dirichlet Series”. The Fibonacci
Quarterly, 48(1) (2010), 47-55.
[5] C. Levesque, “On m-th order linear recurrences”, The Fibonacci Quarterly,
23(4) (1985), 290-293.
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Numbers”, International Journal of Mathematical Analysis, 7(12) (2013),
551-556.
[7] D. Marques, “The Order of Appearance of the Product of Consecutive Lucas
Numbers”, The Fibonacci Quarterly, 51(1) (2013), 38-43.
[8] F. Koken, D. Bozkurt, “On the Jacobsthal numbers by matrix methods”, Int.
J. Contemp. Math. Sciences, 3(13) (2008), 605-614.
Modified k-Pell sequence
6037
[9] F. Yilmaz, D. Bozkurt, “The Generalized Order-k Jacobsthal Numbers”, Int.
J. Contemp. Math. Sciences, 4(34) (2009), 1685-1694.
[10] M. Shattuck, “Combinatorial Proofs of Determinant Formulas for the
Fibonacci and Lucas Polynomials”, The Fibonacci Quarterly, 51(1) (2013),
63-71.
[11] N. N. Vorobiov, Números de Fibonacci, Editora MIR, URSS, 1974.
[12] P. Catarino, “On some identities and generating functions for k-Pell
numbers”, International Journal of Mathematical Analysis, 7(38) (2013),
1877-1884.
[13] P. Catarino, P. Vasco, “Some basic properties and a two-by-two matrix
involving the k-Pell Numbers”, International Journal of Mathematical
Analysis, 7(45) (2013), 2209-2215.
[14] P. Catarino, P. Vasco, “On some Identities and Generating Functions for kPell-Lucas sequence”, Applied Mathematical Sciences, 7(98) (2013),
4867-4873.
[15] S. Falcón, Á. Plaza, “On the Fibonacci k-numbers”, Chaos, Solitons &
Fractals, 32(5) (2007), 1615-1624.
[16] V. E. Hoggatt, Fibonacci and Lucas Numbers. A publication of the
Fibonacci Association. University of Santa Clara, Santa Clara. Houghton
Mifflin Company, 1969.
Received: September 17, 2013