On Some Identities for K-fibonacci Sequence

We obtain some identities for k-Fibonacci numbers by using its Binet's formula. Also, another expression for the general term of the sequence, using the ordinary generating function, is provided.


Introduction
The well-known Fibonacci sequence is one of the sequences of numbers that have been studied over several years.Many papers and research work are dedicated to Fibonacci sequence, such as the work of Hoggatt in [13] and Vorobiov, in [6], among others.Also relating with Fibonacci sequence, in Falcón and Plaza, [12], we find some properties for k-Fibonacci numbers obtained from elementary matrix algebra and its identities including generating function and divisibility properties appears in Bolat and Köse, in [1].The Fibonacci sequence is one of the sequence that is defined recursively as well as the sequences of Pell, Pell-Lucas and Modified Pell numbers.More recently, P. Catarino and P. Vasco did some research about the sequences of numbers that arising from these sequences: for any positive real number k, the k-Pell sequence {P k,n } n , k-Pell-Lucas sequence {Q k,n } n and Modified k-Pell sequence {q k,n } n , that are also defined by recursive recurrences.In these cases, for ݊ 1, we have, respectively, the following: ܲ ,ାଵ ൌ 2ܲ , + kܲ ,ିଵ , ܲ , ൌ 0, ܲ ,ଵ ൌ 1; ܳ ,ାଵ ൌ 2ܳ , + kܳ ,ିଵ , ܳ , ൌ ܳ ,ଵ ൌ 2; ‫ݍ‬ ,ାଵ ൌ ‫ݍ2‬ , + k‫ݍ‬ ,ିଵ , ‫ݍ‬ , = ‫ݍ‬ ,ଵ ൌ 1.For more detail about these sequences, see the works of Catarino [7], [11] and Catarino and Vasco [8], [9] and [10].
The Binet's formula is also well known for several of these sequences.Claude Levesque, in [2] finds the general Binet's formula for a general m th order linear recurrence.We have, ଶ , where ‫ݎ‬ ଵ ൌ 1 √1 ݇ and ‫ݎ‬ ଶ ൌ 1 െ √1 ݇ are the roots of the characteristic equation of the sequences {P k,n } n , {Q k,n } n and {q k,n } n , respectively.As a curiosity, for ݇ ൌ 1, we obtain the silver ratio which is related with the Pell number sequence.Silver ratio is the limiting ratio of consecutive Pell numbers.Sometimes some basic properties come from the Binet's formula.For example, for the sequence of Jacobsthal number, Koken and Bozkurt, in [4], deduce some properties and the Binet's formula, using matrix method.In [5], Yilmaz and Bozkurt study some more properties related with k-Jacobsthal numbers as well as Jhala et al. in [3], Catarino in [7] and Catarino and Vasco in [9] and [10] did a similar study for k-Pell, k-Pell-Lucas and Modified k-Pell sequences and many properties are proved by easy arguments for the k-Pell, k-Pell-Lucas and Modified k-Pell number.In this paper, we consider the k-Fibonacci sequence and many identities are proved by easy arguments for the k-Fibonacci number.
(2) Consider the following characteristic equation, associated to the recurrence relation (1), with two distinct roots ߙ ଵ and ߙ ଶ .Note that the roots of the equation ( 3) are , where ݇ is a real positive number.Since √݇ ଶ 4 1, then ߙ ଶ ൏ 0 and so As a curiosity, for k ൌ 1 we obtain that ߙ ଵ ൌ ଵା √ହ ଶ is the golden number which is related with the Fibonacci number sequence ሼ‫ܨ‬ ሽ ∈ Գ .Golden number is the limiting ratio of consecutive Fibonacci numbers.The following formula was established by Bolat and Köse in [1].
Proposition 2 (Catalan's identity) Proof: Using the Binet's formula (4) and the fact that ߙ ଵ ߙ ଶ ൌ െ1, we get that is, the identity required.∎ Note that for ‫ݎ‬ ൌ 1 in Catalan's identity obtained, we get the Cassini's identity for the ݇-Fibonacci numbers sequence.In fact, the equation ( 5) for ‫ݎ‬ ൌ 1, yields ‫ܨ‬ ,ିଵ ‫ܨ‬ ,ାଵ െ ‫ܨ‬ , ଶ ൌ ሺെ1ሻ ‫ܨ‬ ,ଵ ଶ and using one of the initial conditions of this sequence we proved the following result.

Proposition 4 (d'Ocagne's identity) If m > n then
Proof: Once more, using the Binet's formula (4), the fact that ߙ ଵ ߙ ଶ ൌ െ1 and ݉ ݊, we get Again using the Binet's formula (4) we obtain other property of the ݇-Fibonacci number sequence which is stated in the following proposition.
Proof: We have that Using the ratio Next we use this fact writing ( 9) with an equivalent form using this ratio, obtaining ∎ Also, we easily can show the following result using basic tools of calculus of limits and the (8).

Generating functions for the k-Fibonacci sequences
The generating functions for the k-Fibonacci sequences are given by Bolat and Röse in Proposition13 in [1].As we know the k-Fibonacci sequences (and other sequences) can be considered as the coefficients of the power series of the corresponding generating function.Bolat and Röse in this Proposition listed some properties of the generating functions for the k-Fibonacci sequences.One of the equalities given in [1] is the following Recall that for a sequence ሺܽ ሻ , if lim →ஶ శభ ൌ ‫,ܮ‬ where is a positive real number, then, considering the power series ∑ ܽ ‫ݔ‬ ஶ ୀ , its radius of convergence ܴ is equal to ଵ .So, for the ݇-Fibonacci sequence, using (8) and then (10) we know that she can be written as a power series with radius of convergence equal to ቂ.Then we have that where ݂ ሺሻ ሺ‫ݔ‬ሻ denotes the nth order derivative of the function ݂.
Proof: We have from (11)