1. Introduction

The Fibonacci sequence and the Lucas sequence are the two shining stars in the vast array of integer sequences. They have fascinated both amateurs and professional mathematicians for centuries, and they continue to charm us with their beauty, abundant applications, and ubiquitous habit of occurring in totally surprising and unrelated places.

The Fibonacci numbers are terms of the sequence {0, 1, 1, 2, 3, 5, 8, 13, 21,. ..}, where each term is the sum of the two previous terms, beginning with the values. The Lucas numbers are terms of the sequence {2, 1, 3, 4, 7, 11, 18, 29, 47,...}, where each term is the sum of the two previous terms, beginning with the values. Many kinds of generalizations of Fibonacci numbers and Lucas numbers have been presented in the literature ^{[6, 7, 9]}. One example of this recent generalization is k-Fibonacci numbers ^[2] and k-Lucas numbers ^[4], which are defined by the recurrence relation, depend on one real parameter k. Falcon and Plaza ^[3] defined k- Fibonacci hyperbolic functions and deduced some properties of k- Fibonacci hyperbolic functions related with the analogous identities for the k- Fibonacci numbers. Falcon ^[4] studied the k- Lucas numbers and proved various properties related with the k- Fibonacci numbers. Thongmoon ^[11] and Hoggatt ^[5] defined various identities for the Fibonacci and Lucas numbers. Gupta and Panwar ^[10], present generalized identities involving common factors of generalized Fibonacci, Jacobsthal and jacobsthal-Lucas numbers and Binet’s formula will employ to obtain the identities. There are a lot of identities of Fibonacci and Lucas numbers described in ^{[1, 8]}. In this paper, we present generalized identities involving common factors of k- Fibonacci, and k-Lucas numbers.

2. The K-Fibonacci and the K-Lucas Numbers

Before presenting our main results, we will need to introduce some known results and notations.

The k-Fibonacci numbers have been defined in ^[2] for any real k as follows:

For any positive real number k, the k-Fibonacci sequence say is define recurrently by

(1)

With initial condition

(2)

Particular cases of the k-Fibonacci sequence are

If k = 1, the classical Fibonacci sequence is obtained

If k = 2, the classical Pell sequence is obtained

If k = 3, the following sequence is obtained

Binet formula for the n^th k-Fibonacci numbers is given by

(3)

where , are the roots of the characteristic equation and ;, which gives

(4)

Similarly for any positive real number k, the k-Lucas sequence ^[4] say is define recurrently by

(5)

with initial condition

(6)

Particular cases of the k-Lucas sequence are

If k = 1, the classical Lucas sequence is obtained

If k = 2, the classical Pell- Lucas sequence is obtained

If k = 3, the following sequence is obtained

Binet formula for the n^th k-Lucas numbers is given by

(7)

where , are the roots of the characteristic equation which are given in equation (4).

There are a lot of identities for the common factor of Fibonacci and the Lucas numbers define in ^[12].

3. Identities for The Common Factors of K-Fibonacci and K-Lucas Number

In this section we present generalized identities involving common factors of k-Fibonacci and k-Lucas numbers. We shall use the Binet’s formula for the k-Fibonacci numbers and k-Lucas numbers for derivation.

Proof

(8)

Proof

(9)

Proof

(10)

Proof

(11)

Proof

(12)

Proof

(13)

Proof

(14)

In ^[13], if we take in Lemma 1(i), then we have the following proposition.

Proof

(15)

Proof

(16)

In ^[13], if we take in Lemma 1(ii), then we have the following proposition.

Proof

(17)

For different values of p, Proposition (3.1) to Proposition (3.10) can be expressed for even and odd k-Fibonacci and k-Lucas numbers.

Identities (8) to (17) give a relation between k-Fibonacci and k-Lucas numbers by using their subscripts.

4. Conclusion

In this paper we present generalized identities involving common factors of k- Fibonacci and k-Lucas numbers. Mainly Binet’s formula employ for the identities. The concept can be executed for generalized second order recursive sequences as well as polynomials.

References

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