On the Products of k-Fibonacci Numbers and k-Lucas Numbers

Fibonacci numbers possess wonderful and amazing properties; though some are simple and known, others find broad scope in research work. Fibonacci and Lucas numbers cover a wide range of interest in modern mathematics as they appear in the comprehensive works of Koshy [1] and Vajda [2]. The Fibonacci numbers F n are the terms of the sequence {0, 1, 1, 2, 3, 5, 8 ⋅ ⋅ ⋅ } wherein each term is the sum of the two previous terms beginning with the initial values F 0 = 0 and


Introduction
Fibonacci numbers possess wonderful and amazing properties; though some are simple and known, others find broad scope in research work. Fibonacci and Lucas numbers cover a wide range of interest in modern mathematics as they appear in the comprehensive works of Koshy [1] and Vajda [2]. The Fibonacci numbers are the terms of the sequence {0, 1, 1, 2, 3, 5, 8 ⋅ ⋅ ⋅ } wherein each term is the sum of the two previous terms beginning with the initial values 0 = 0 and 1 = 1. Also the ratio of two consecutive Fibonacci numbers converges to the Golden mean, 0 = (1 + √ 5)/2. The Fibonacci numbers and Golden mean find numerous applications in modern science and have been extensively used in number theory, applied mathematics, physics, computer science, and biology.
The well-known Fibonacci sequence is defined as In a similar way, Lucas sequence is defined as The second order Fibonacci sequence has been generalized in several ways. Some authors have preserved the recurrence relation and altered the first two terms of the sequence while others have preserved the first two terms of the sequence and altered the recurrence relation slightly.
Motivated by the study of -Fibonacci numbers in [4], the -Lucas numbers have been defined in a similar fashion as 2 International Journal of Mathematics and Mathematical Sciences The first few terms of this sequence are The particular cases of the -Lucas sequence are as follows. If = 1, the classical Lucas sequence is obtained: If = 2, the Pell-Lucas sequence is obtained: {2, 2, 6, 14, 34, 82 ⋅ ⋅ ⋅ } .
In the 19th century, the French mathematician Binet devised two remarkable analytical formulas for the Fibonacci and Lucas numbers [2]. The same idea has been used to develop Binet formulas for other recursive sequences as well. The wellknown Binet's formulas for -Fibonacci numbers and -Lucas numbers, see [3][4][5], are given by where 1 , 2 are roots of characteristic equation which are given by We also note that There are a huge number of simple as well as generalized identities available in the Fibonacci related literature in various forms. Some properties for common factors of Fibonacci and Lucas numbers are studied by Thongmoon [6,7]. The -Fibonacci numbers which are of recent origin were found by studying the recursive application of two geometrical transformations used in the well-known fourtriangle longest-edge partition [3], serving as an example between geometry and numbers. Also in [8], authors established some new properties of -Fibonacci numbers and -Lucas numbers in terms of binomial sums. Falcón and Plaza [9] studied 3-dimensional -Fibonacci spirals considering geometric point of view. Some identities for -Lucas numbers may be found in [9]. In [10] many properties of -Fibonacci numbers are obtained by easy arguments and related with so-called Pascal triangle. The aim of the present paper is to establish connection formulas between -Fibonacci and -Lucas numbers, thereby deriving some results out of them. In the following section we investigate some products of -Fibonacci numbers and -Lucas numbers. Though the results can be established by induction method as well, Binet's formula is mainly used to prove all of them.

Proof.
,2 +1 ,2 In the same manner, we obtain the following results.

Proof.
, , For different value of , we have different results: Similarly we have the following result.