1. Introduction
Let us remember the k-Lucas numbers Lk,n are defined [1] by the recurrence relation 
with the initial conditions 
Among other properties, the Binnet Identity establishes 
being 
and 
the characteristic roots of the recurrence equation
.
Evidently,
.
Moreover, it is verified [1, Theorem 2.4] that
.
If we apply iteratively the equation 
then we will find a formula that relates the k–Lucas numbers to the k–Fibonacci numbers:
(1.1)
This formula is similar to the Convolution formula for the k–Fibonacci numbers 
[2,3].
Moreover, we define
. Then, if we do p = −n in Formula (1.1) obtain
.
2. On the k-Lucas Numbers of Arithmetic Index
We begin this section with a formula that relates each other some k-Lucas numbers.
2.1. Theorem 1 (The k-Lucas Numbers of Arithmetic Index)
If a is a nonnull natural number and r = 0, 1, 2, ... a − 1, then
(2.1)
Proof. In [4] it is proved
.
Then
If r = 0, then 
In this case, if a = 2p + 1, then an odd k-Lucas number can be expressed in the form
Applying iteratively Formula (2.1), the general term, for
, can be written like a non-linear combination of the form 
In particular, if m = n, then
2.2. Generating Function of the Sequence {Lk, an + r}
Let 
be the generating function of the sequence
. That is,
Then,
and
from where 
no more to take into account Formula (2.1). So, the generating function of the sequence 
is
.
As particular case, if a = 1, then r = 0 and the generating function of the k-Lucas sequence 
is 
, that, for the classical Lucas sequence is 
If we want to take out the two bisection sequences of the classical Lucas sequence (k = 1), the respective generating functions are a = 2 and r = 0: 
that generates the sequence 
a = 2 and r = 1: 
that generates the sequence
.
2.3. Theorem 2 (Sum of the k-Lucas Numbers of Arithmetic Index)
If a is a nonnull natural number and r = 0, 1, 2, ... a − 1, then
(2.2)
Proof.
because
and after applying the formula for the sum of a geometric progression.
2.4. Corollary 1 (Sum of Consecutive Odd k-Lucas Numbers)
If r = 0 and a = 2p + 1, Equation (2.2) is
In this case, the sum of the first k-Lucas numbers is (for p = 0),
(2.3)
that for the classical Lucas numbers is 
2.5. Corollary 2 (Sum of Consecutive Even k-Lucas Numbers)
If r = 0 and a = 2p, then Equation (2.2) is
(2.4)
In this case, if p = 1 we obtain the formula for the sum of the first even k-Lucas numbers
, and for the classical Lucas numbers is 
2.6. Theorem 3 (Sum of Alternated k-Lucas Numbers of Arithmetic Index)
For a > 0 and r = 0, 1, 2, ...a − 1, the sum of alternated k-Lucas numbers is
Proof. As in the previous theorem,
2.7. Corollary 3 (Sum of Consecutive Alternated Odd k-Lucas Numbers)
As particular case, if a = 2p + 1 and r = 0,
Then, for p = 0 we obtain the sum of the first alternated k-Lucas numbers
, that for the classical Lucas numbers is
.
2.8. Corollary 4 (Sum of Consecutive Alternated Even k-Lucas Numbers)
If r = 0 and a = 2p + 1, then
And for the first consecutive alternated even k-Lucas numbers
that for the classical Lucas numbers is
.
3. On the k-Fibonacci Numbers of Indexes n and the k-Lucas Numbers
In this section we will study a relation between the numbers 
and
.
3.1. Theorem 4 (A Relation between Some k-Fibonacci and the k-Lucas Numbers)
For r ≥ 1, it is
(3.1)
Proof.
In particular, if r = 1, it is 
Taking into account
, if we expand Formula (3.1), we find that this formula can be expressed as 
or, that is the same,
Then, applying Formula (2.2) to the second hand right of this equation with
, a = 4n, and r = 3n for the first term and r = n for the second,
(3.2)
We tray to simplify the second hand right of this equation. For that, we will prove the following Lemma.
3.2. Lemma 1
(3)
Proof. We will apply the following formulas:
(relation)
(negative)
(convolution)
(definition)
Then:
(by relation)
(by convolution)
(by negative)
(by definition)
And applying this Lemma to Equation (3.2), we will have:
that is
from where
If in Equation (3.3) it is a = 0, then it is 
, and applying the Formulas (2.5) and (2.4),
That is
In particular, for the classical Lucas numbers (k = 1), it is
.
4. Acknowledgements
This work has been supported in part by CICYT Project number MTM200805866-C03-02 from Ministerio de Educación y Ciencia of Spain.