Keywords: Generalized Fibonacci numbers, kGeneralized Fibonacci numbers, Binet’s formula
Applied Mathematics and Physics, 2014 2 (1),
pp 1012.
DOI: 10.12691/amp213
Received November 30, 2013; Revised December 21, 2013; Accepted January 08, 2014
Copyright © 2014 Science and Education Publishing. All Rights Reserved.
1. Introduction
It is wellknown that the Fibonacci sequence is most prominent examples of recursive sequence. The Fibonacci sequence is famous for possessing wonderful and amazing properties. The Fibonacci appear in numerous mathematical problems. Fibonacci composed a number text in which he did important work in number theory and the solution of algebraic equations. Many authors have generalized second order recurrence sequences by preserving the recurrence relation and alternating the first two terms of the sequence and some authors have generalized these sequences by preserving the first two terms of the sequence but altering the recurrence relation slightly.
The kFibonacci numbers defined by Falco’n and Plaza ^{[6]}, depending only on one integer parameter k as follows,
For any positive real number k, the kFibonacci sequence is defined recurrently by
 (1.1) 
.
Many of the properties of these sequences are proved by simple matrix algebra. This study has been motivated by the arising of two complex valued maps to represent the two antecedents in an specific fourtriangle partition. In ^{[8]}, Falcon and Plaza k Fibonacci sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and related with the socalled Pascal 2triangle. New generalized kFibonacci sequences have been introduced and studied. Several properties of these numbers are deduced and related with the socalled Pascal 2triangle. In addition, the generating functions for these kFibonacci sequences have been given.
The Generalized Fibonacci sequences ^{[12]}, defined by Gupta, Panwar and Sikhwal, as follows,
 (1.2) 
.
If , we get
 (1.3) 
.
The first few terms of are 2, 0, 4, 4, 12, 20 and so on.
In this paper, we introduced the kGeneralized Fibonacci sequence. Also we establish some of the interesting properties of kGeneralized Fibonacci numbers. In this paper, we present properties of kGeneralized Fibonacci numbers like Catalan’s identity, Cassini’s identity and d’ocagnes’s Identity.
2. The kGeneralized Fibonacci Numbers
In this section, we define kgeneralized Fibonacci sequence. For , the kgeneralized Fibonacci sequence is defined by
 (2.1) 
.
The first few kgeneralized Fibonacci numbers are
Particular case of kgeneralized Fibonacci number is
If generalized Fibonacci sequence is obtained
3. Properties of kGeneralized Fibonacci Numbers
3.1. First Explicit Formula for kGeneralized Fibonacci Numbers In the 19th century, the French mathematician Binet devised two remarkable analytical formulas for the Fibonacci and Lucas numbers. In our case, Binet’s formula allows us to express the generalized kFibonacci numbers in function of the roots of the following characteristic equation, associated to the recurrence relation (2.1)
 (3.1) 
Theorem 1: (Binet’s formula). The nth kgeneralized Fibonacci number is given by
 (3.2) 
where are the roots of the characteristic equation (3.1) and .
Proof: We use the Principle of Mathematical Induction (PMI) on n. It is clear the result is true for by hypothesis. Assume that it is true for r such that , then
It follows from definition of kgeneralized Fibonacci numbers (2.1) and equation (3.2)
Thus, the formula is true for any positive integer n.
where and .
This completes the proof.
3.2. Catalan's IdentityCatalan's identity for Fibonacci numbers was found in 1879 by Eugene Charles Catalan a Belgian mathematician who worked for the Belgian Academy of Science in the field of number theory.
Theorem 2: (Catalan’s identity)
 (3.3) 
Proof: By Binet’s formula (3.2), we have
This completes the Proof.
3.3. Cassini's IdentityThis is one of the oldest identities involving the Fibonacci numbers. It was discovered in 1680 by JeanDominique Cassini a French astronomer.
Theorem 3: (Cassini’s identity or Simpson’s identity)
 (3.4) 
Proof. Taking in Catalan’s identity (3.3) the proof is completed.
3.4. D'Ocagne's IdentityTheorem 4: (d’ocagnes’s Identity)
 (3.5) 
Proof. By Binet’s formula (3.2), we have
This completes the Proof.
3.5. Limit of the Quotient of Two Consecutive TermsA useful property in these sequences is that the limit of the quotient of two consecutive terms is equal to the positive root of the corresponding characteristic equation
Theorem 5:
 (3.6) 
Proof. By Binet’s formula (3.2), we have
and taking into account that , since , Eq. (3.6) is obtained.
3.6. Generating function for the kGeneralized Fibonacci Sequence:Generating functions provide a powerful technique for solving linear homogeneous recurrence relations. Even though generating functions are typically used in conjunction with linear recurrence relations with constant coefficients, we will systematically make use of them for linear recurrence relations with non constant coefficients. In this paragraph, the generating function for kgeneralized Fibonacci sequence is given. As a result, kgeneralized Fibonacci sequence is seen as the coefficients of the corresponding generating function. Function defined in such a way is called the generating function of the kgeneralized Fibonacci sequence. So,
and then,
 (3.7) 
4. Conclusion
In this paper new kgeneralized Fibonacci sequence have been introduced and studied. Many of the properties of this numbers are proved by simple algebra and Binet’s formula. Finally we present properties like Catalan’s identity, Cassini’s identity or Simpson’s identity and d’ocagnes’s identity for kgeneralized Fibonacci numbers.
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