Some Identities Involving the Generalized Lucas Numbers

Received: August 03, 2020 Revised: September 10, 2020 Accepted: September 16, 2020 Published Online: October 19, 2020 The terms of the sequence of Lucas numbers Ln { } can be obtained by L L 0 1 2 1 = = , and the recurrence relation L L L n n n = + − − 1 2 ; for n≥ 2 . There are several ways in which it has been generalized. One of these ways is by preserving the initial conditions and changing the recurrence relation. Whereas one more way is to preserve the recurrence relation and alternate the initial conditions. One of the generalizations of the Lucas sequence is the class of sequences Gn L a b , ( ) { } generated by the recurrence relation.

2 ; for n ≥ 2 . There are several ways in which it has been generalized. One of these ways is by preserving the initial conditions and changing the recurrence relation. Whereas one more way is to preserve the recurrence relation and alternate the initial conditions. One of the generalizations of the Lucas sequence is the class of sequences G n   ( ) to compute the approximate value of its successor and predecessor. We also establish some amusing identities for this sequence displaying the relation between G n

Introduction
In the theory of numbers, the Fibonacci sequence has been always fertile ground for the mathematicians. At the same time, the Lucas sequence, being its twin sequence, also has this nature. . In recent years, many interesting properties of Fibonacci numbers , Lucas numbers and their generalizations have been shown by researchers and applied to almost every field of science and art. Bacani et al 2015, Falcon 2014Gupta et al., 2012 have generalized the sequence of Fibonacci numbers in different ways and obtain many of the interesting results. Bolat et al., 2013, Kaygisiz et al., 2012and Shah et al., 2015 defined new generalizations of Lucas sequence and gave various identities along with extended Binet formula for the concerned new generalizations. In this paper we further generalize these sequences and introduce generalized Lucas numbers as follows: Definition: For any two positive integers a and b, the generalized Lucas sequence is defined by G  if n is odd ; . if n is even where n 2 Some initial terms of this sequence are Clearly that G L n L n 1 1 , ( ) = . In this paper, using the techniques of generating functions, we derive the extended Binet formula for this generalized Lucas numbers and develop some interesting results for them. For convenience, throughout the paper, we use G n for G n (1) αβ = −ab Mathematician Jacques Philippe Marie Binet derived the Binet's formula, which is used to find the n th term of the Fibonacci sequence. The formula states Similar explicit formulae areknown for the various generalizations of Fibonacci number. In this section, we fist obtain explicit Binet -type formula for G n .
. These gives Using the definition of G n , we get 1 2 2 1 3 3 Then this gives Then by (1)  , as required.
We next use this formula obtained to express G n as an infinite series.
Case: 1 If we consider n = 2k then it follows that   First we consider ab ≤ 4. In this case for n = 2, we get the following values.

Summary
In this article, we have defined the new class of generalized Lucas sequence G n L a b , ( ) { } and obtained the extended Binet formula for it. We have also obtained recursive formula for G n L a b , ( ) to obtain the approximate values of its successor and predecessor. We have also shown the relation of this sequence with classical Fibonacci sequence and classical Lucas sequence.

Authorship Contribution
This work is done by myself under the guidance of my Ph.D. mentor Dr. Devbhadra V. Shah, who is also the co-author of this paper.

Funding
This work is not funded by any project agency.