Int. Journal of Math. Analysis, Vol

Int. Journal of Math. Analysis, Vol. 7, 2013, no. 38, 1877 - 1884
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijma.2013.35131
On Some Identities and Generating Functions

for k- Pell Numbers
Paula Catarino
1
Department of Mathematics
School of Science and Technology
University of Trás-os-Montes e Alto Douro (Vila Real – Portugal)
[e-mail]
Copyright © 2013 Paula Catarino. 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
We obtain the Binet’s formula for k-Pell numbers and as a consequence we
get some properties for k-Pell numbers. Also we give the generating function for
k-Pell sequences and another expression for the general term of the sequence, using
the ordinary generating function, is provided.
Mathematics Subject Classification: 11B37, 05A15, 11B83.
Keywords: Pell numbers, k-Pell numbers, Generating functions, Binet’s formula
1. Introduction
Some sequences of number have been studied over several years, with
emphasis on studies of well-known Fibonacci sequence (and then the Lucas
sequence) that is related to the golden ratio. Many papers and research work are
dedicated to Fibonacci sequence, such as the work of Hoggatt, in [15] and
Vorobiov, in [13], among others and more recently we have, for example, the
works of Caldwell et al. in [4], Marques in [7], and Shattuck, in [11]. Also

1
Collaborator of CIDMA and CM-UTAD (Portuguese researches centers). 1878 Paula Catarino
relating with Fibonacci sequence, Falcın et al., in [14], consider some properties
for 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]. Other sequence, also important, is the sequence of Pell
numbers defined by the recursive recurrence given by
  2
 
 ,

 2, with
  0 and
  1. This sequence has been studied and some its
basic properties are known (see, for example, the study of Horadam, in [2]). In [10],
we find the matrix method for generating this sequence and comparable matrix
generators have been considered by Kalman, in [6], by Bicknell, in [12], for the
Fibonacci and Pell sequences. From this sequence, we obtain some types of other
sequences namely, Pell-Lucas and Modified Pell sequences and also Dasdemir, in
[1], consider some new matrices which are based on these sequences as well as
that they have the generating matrices. The Binet’s formula is also well known for
several of these sequences. Sometimes some basic properties come from this
formula. For example, for the sequence of Jacobsthal number, Koken and Bozkurt,
in [8], deduce some properties and the Binet’s formula, using matrix method. In
[9], Yilmaz et al. study some more properties related with k- Jacobsthal numbers.
According Jhala et al., in [5], we consider, in this paper, the k-Pell numbers
sequence and many properties are proved by easy arguments for the k-Pell
number.
2. The k-Pell Number and some identities

For any positive real number k, the k-Pell sequence say 
, ∈  is
defined recurrently by

,  2
, + k
,, for
 1, (1)
with initial conditions given by,

,  0,
,  1. (2)
Next we find the explicit formula for the term of order n of the k-Pell numbers
sequence using the well-known results involving recursive recurrences. Consider
the following characteristic equation, associated to the recurrence relation (1),

 2    0, (3)
with two distinct roots  and  . Note that the roots of the equation (3) are
  1  √1   and   1  √1   , where  is a real positive number.
Since√1    1, then   0 and so,   0  .
Also, we obtain that     2,     2√1   and   .
As a curiosity, for k  1, we obtain that   1  √2 is the silver ratio which is
related with the Pell number sequence. Silver ratio is the limiting ratio of
consecutive Pell numbers. Identities and generating functions for k-Pell numbers 1879
Proposition 1 (Binet’s formula)
The nth k-Pell number is given by

, 


 

 
(4)
where ,  are the roots of the characteristic equation (3) and  >  .
Proof: Since the equation (3) has two distinct roots, the sequence

, = c1(r1)
n
+ c2(r2)
n
(5)
is the solution of the equation (1). Giving to n the values n = 0 and n = 1 and
solving this system of linear equations, we obtain a unique value for c1 and c2. So,
we get the following distinct values, c1 =

√
and c2 = 

√
=  c1.
Since     2√1  , we can express c1 and c2, respectively by c1 =

 
and c2 = 

 
. Now, using (5), we obtain (4) as required.
∎
Proposition 2 (Catalan’s identity)

,
, 
,

 1




,

(6)
Proof: Using the Binet’s formula (4) and the fact that   , we get

,
, 
,

 !

"#
 
"#

$ !

%#
 
%#

$  !


 


$

 1



!

#

#

$

 1




,

,
that is, the identity required. ∎
Note that for   1 in Catalan’s identity obtained, we get the Cassini’s identity
for the -Pell numbers sequence. In fact, the equation (6), for   1, yields

,
, 
,

 1




,

and using one of the initial conditions of this sequence, we proved the following
result.
Proposition 3 (Cassini’s identity)

,
, 
,

 1



(7)
∎ 1880 Paula Catarino
The d’Ocagne’s identity can also obtained using the Binet’s formula as it was done
by Jhala et al. in [5] for the k-Jacobsthal sequence. Hence we have
Proposition 4 (d’Ocagne’s identity)
If m > n then
,&
, 
,&
,  1




,&. (8)
Proof: Once more, using the Binet’s formula (4), the fact that    and
' 
, we get

,
, 
,

 !

( 
(

$ !

%
 
%

$  !

(%
 
(%

$ !


 


$
 


!


("
 
("



$
 


!

("
 
("

$
 


,&
 1




,&.
∎
Again using the Binet’s formula (4) we obtain other property of the  -Pell
sequence which is stated in the following proposition.
Proposition 5
lim→-
./,
./,"
 . (9)
Proof: We have that
lim→-
./,
./,"
 lim→- !


 


$ !



"
 

"$  lim→- !


 



"
 

"$. (10)
Using the ratio


and since 0


0  1, then lim→- !


$

 0. Next we use this
fact writing (10) with an equivalent form using this ratio, obtaining
lim→-
./,
./,"
 lim→-
!
#
#
$


#
!
#
#
$
 
#
 lim→-


#
 .
∎
Also, we easily can show the following result using basic tools of calculus of limits
and the (9). Identities and generating functions for k-Pell numbers 1881
Proposition 6
lim→-
./,"
./,



. (11)
∎
3. Generating functions for the k-Pell sequences
Next we shall give the generating functions for the k-Pell sequences. We shall
write the k-Pell sequence as a power series where each term of the sequence
correspond to coefficients of the series and from that fact, we find the generating
function. Let us consider the k-Pell sequences 1
,2

for any positive integer k.
By definition of ordinary generating function of some sequence, considering this
sequence, the ordinary generating function associated is defined by
3
,; 5  ∑
,5
- 
7 
, 
,5 
, 5
 ⋯ 
,5

+… (12)
Using the initial conditions, we get
3
,; 5  5  ∑
,5
- 
7
. (13)
Now from (1) we can write (13) as follows
3
,; 5  5  ∑ 2
,  
, 5
- 
7
. (14)
Consider the right side of the equation (14) and doing some calculations, we
obtain that
5  92
,  
, 5

-
7
 5  29
,5

-
7
  9
, 5

-
7
 5  25 ∑
,5
  5
-
7 ∑
, 5
- 
7
. (15)
Consider that : 
 2 and ; 
 1. Then (15) can be written by
5  25 9
,

ผลลัพธ์ (ไทย) 1:

Int. Journal of Math. Analysis, Vol. 7, 2013, no. 38, 1877 - 1884 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.35131 On Some Identities and Generating Functions for k- Pell Numbers Paula Catarino1Department of Mathematics School of Science and Technology University of Trás-os-Montes e Alto Douro (Vila Real – Portugal) [e-mail] Copyright © 2013 Paula Catarino. 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 We obtain the Binet’s formula for k-Pell numbers and as a consequence we get some properties for k-Pell numbers. Also we give the generating function for k-Pell sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided. Mathematics Subject Classification: 11B37, 05A15, 11B83. Keywords: Pell numbers, k-Pell numbers, Generating functions, Binet’s formula1. Introduction Some sequences of number have been studied over several years, with emphasis on studies of well-known Fibonacci sequence (and then the Lucas sequence) that is related to the golden ratio. Many papers and research work are dedicated to Fibonacci sequence, such as the work of Hoggatt, in [15] and Vorobiov, in [13], among others and more recently we have, for example, the works of Caldwell et al. in [4], Marques in [7], and Shattuck, in [11]. Also 1 Collaborator of CIDMA and CM-UTAD (Portuguese researches centers). 1878 Paula Catarinorelating with Fibonacci sequence, Falcın et al., in [14], consider some properties for k-Fibonacci numbers obtained from elementary matrix algebra and itsidentities including generating function and divisibility properties appears in the paper of Bolat et al., in [3]. Other sequence, also important, is the sequence of Pell numbers defined by the recursive recurrence given by
  2
 
 , 2, with
 0 and
  1. This sequence has been studied and some its basic properties are known (see, for example, the study of Horadam, in [2]). In [10], we find the matrix method for generating this sequence and comparable matrix generators have been considered by Kalman, in [6], by Bicknell, in [12], for the Fibonacci and Pell sequences. From this sequence, we obtain some types of other sequences namely, Pell-Lucas and Modified Pell sequences and also Dasdemir, in [1], consider some new matrices which are based on these sequences as well as that they have the generating matrices. The Binet’s formula is also well known for several of these sequences. Sometimes some basic properties come from this formula. For example, for the sequence of Jacobsthal number, Koken and Bozkurt, in [8], deduce some properties and the Binet’s formula, using matrix method. In [9], Yilmaz et al. study some more properties related with k- Jacobsthal numbers. According Jhala et al., in [5], we consider, in this paper, the k-Pell numbers sequence and many properties are proved by easy arguments for the k-Pell number. 2. The k-Pell Number and some identities For any positive real number k, the k-Pell sequence say 
, ∈  is defined recurrently by
,  2
, + k
,, for 1, (1) with initial conditions given by,
,  0,
,  1. (2) Next we find the explicit formula for the term of order n of the k-Pell numbers sequence using the well-known results involving recursive recurrences. Consider the following characteristic equation, associated to the recurrence relation (1),   2    0, (3) with two distinct roots  and  . Note that the roots of the equation (3) are   1  √1   and   1  √1   , where  is a real positive number. Since√1    1, then   0 and so,   0  . Also, we obtain that     2,     2√1   and   . As a curiosity, for k  1, we obtain that   1  √2 is the silver ratio which is related with the Pell number sequence. Silver ratio is the limiting ratio of consecutive Pell numbers. Identities and generating functions for k-Pell numbers 1879 Proposition 1 (Binet’s formula) The nth k-Pell number is given by
,     (4) where ,  are the roots of the characteristic equation (3) and  >  . Proof: Since the equation (3) has two distinct roots, the sequence
, = c1(r1)n + c2(r2)n (5) is the solution of the equation (1). Giving to n the values n = 0 and n = 1 and solving this system of linear equations, we obtain a unique value for c1 and c2. So, we get the following distinct values, c1 =  √ and c2 =  √ =  c1. Since     2√1  , we can express c1 and c2, respectively by c1 =  and c2 =  . Now, using (5), we obtain (4) as required. ∎ Proposition 2 (Catalan’s identity)
,
, 
,  1
, (6) Proof: Using the Binet’s formula (4) and the fact that   , we get
,
, 
,  !"# "#$ !%# %#$  ! $  1 !##$  1 
, , that is, the identity required. ∎ Note that for   1 in Catalan’s identity obtained, we get the Cassini’s identity for the -Pell numbers sequence. In fact, the equation (6), for   1, yields
,
, 
,  1
, and using one of the initial conditions of this sequence, we proved the following result. Proposition 3 (Cassini’s identity)
,
, 
,  1 (7) ∎ 1880 Paula CatarinoThe d’Ocagne’s identity can also obtained using the Binet’s formula as it was done by Jhala et al. in [5] for the k-Jacobsthal sequence. Hence we have Proposition 4 (d’Ocagne’s identity) If m > n then
,&
, 
,&
,  1
,&. (8) Proof: Once more, using the Binet’s formula (4), the fact that    and '  , we get
,
, 
,  !( ($ !% %$  !(% (%$ ! $   !(" ("$   !(" ("$  
,&  1
,&. ∎ Again using the Binet’s formula (4) we obtain other property of the  -Pell sequence which is stated in the following proposition. Proposition 5 lim→-./,./," . (9) Proof: We have that lim→-./,./," lim→- ! $ !" "$  lim→- ! " "$. (10) Using the ratio  and since 00  1, then lim→- !$ 0. Next we use this fact writing (10) with an equivalent form using this ratio, obtaining lim→-./,./," lim→-!##$#!##$ # lim→-# . ∎ Also, we easily can show the following result using basic tools of calculus of limits and the (9). Identities and generating functions for k-Pell numbers 1881 Proposition 6 lim→-./,"./, . (11) ∎ 3. Generating functions for the k-Pell sequencesNext we shall give the generating functions for the k-Pell sequences. We shall write the k-Pell sequence as a power series where each term of the sequence correspond to coefficients of the series and from that fact, we find the generating function. Let us consider the k-Pell sequences 1
,2 for any positive integer k. By definition of ordinary generating function of some sequence, considering this sequence, the ordinary generating function associated is defined by 3
,; 5  ∑
,5- 7 
, 
,5 
, 5  ⋯ 
,5+… (12) Using the initial conditions, we get 3
,; 5  5  ∑
,5- 7 . (13) Now from (1) we can write (13) as follows 3
,; 5  5  ∑ 2
,  
, 5- 7 . (14) Consider the right side of the equation (14) and doing some calculations, we obtain that 5  92
,  
, 5-7  5  29
,5
-
7
  9
, 5

-
7
 5  25 ∑
,5
  5
-
7 ∑
, 5
- 
7
. (15)
Consider that : 
 2 and ; 
 1. Then (15) can be written by
5  25 9
,<5
<  1  5

-
<7
9
,=5
=
-
=7
 5  25 9
,<5
<  5

-
<7
9
,=5
=  25
-
=71882 Paula Catarino
 5  25 ∑
,<5
<  5
-
<7 ∑
,=5
- =
=7
. (16)
Therefore,
∑
,5
- 
7  5  25 ∑
,5
  5
-
7 ∑
,5
- 
7
,
which is equivalent to
∑
,5
- 
7
1  25  5

  5,
and then the ordinary generating function of the k-Pell sequence can be written as
∑
,5
- 
7 
>
 >>
. (17)
Recall that for a sequence ?

, if lim→-
@%
@
 A, where A is a positive real
number, then, considering the power series ∑ ?5
- 
7
, its radius of convergence
B i