On the complex k-Fibonacci numbers

Abstract: We first study the relationship between the k-Fibonacci numbers and the elements of a subset of Q. Later, and since generally studies that are made on the Fibonacci sequences consider that these numbers are integers, in this article, we study the possibility that the index of the k-Fibonacci number is fractional; concretely, 2n+1 2 . In this way, the k-Fibonacci numbers that we obtain are complex. And in our desire to find integer sequences, we consider the sequences obtained from the moduli of these numbers. In this process, we obtain several integer sequences, some of which are indexed in The Online Enciplopedy of Integer Sequences (OEIS).

ABOUT THE AUTHOR Sergio Falcon Santana obtained his PhD in Mathematics from ULPGC. He is a professor in the University of Las Palmas de Gran Canaria in Mathematics, Advanced Calculus, Numerical Calculus, Algebra...
He is an author of several books on mathematics for technical schools.
Despite getting only about 12 years, dedicated to the University on time complete, at this time, he has published more than 70 articles in mathematics covering various aspects such as disclosure mathematics mathematical didactics and, above all, mathematical research of top level. Some of his research articles have been published in the best mathematics journals around the world.
He has some numerical successions indexed in The On-Line Encyclopedia of Integer Sequences of N.J.A. Sloane and have helped to improve some aspects of others.
He is a partner of the Real Sociedad Matemática Española (RSME) and the Association of teachers of Mathematics "Puig Adam".
He is a member of the University Institute of Applied Microelectronics.

PUBLIC INTEREST STATEMENT
So far, the Fibonacci numbers have been generalized in different forms one being that made by ourselves and we call k-Fibonacci numbers.
But all these generalizations have been in the real numbers field.
What I propose in this study is to extend the area of Fibonacci numbers to the complex numbers field.

On the k-Fibonacci numbers
For any positive real number k, the k-Fibonacci sequence, say F k,n n∈ℕ , is defined recurrently by F k,n+1 = k F k,n + F k,n−1 for n ≥ 1 with initial conditions F k,0 = 0 and F k,1 = 1.
For k = 1, the classical Fibonacci sequence is obtained and for k = 2, the Pell sequence appears.
The well-known Binet formula (Falcon & Plaza, 2007a;Horadam, 1961;Spinadel, 2002) allows us to relate the k-Fibonacci numbers to the characteristic roots 1 and 2 associated to the recurrence If denotes the positive characteristic root, = k+ √ k 2 +4 2 , the general term may be written as F k,n = n − (− ) −n + −1 , and it is verified that the limit of the quotient of two terms of the sequence {F k,n } n∈ℕ is lim n→∞ In particular, if k = 1, then is the Golden Ratio, = 1+ √ 5 2 ; if k = 2, 2 is the Silver Ratio and for k = 3, we obtain the Bronze Ratio (Spinadel, 2002).
Among other properties that we can see in Falcon and Plaza (2007a, 2007b, 2009a, we will need the Simson Identity: F k,n−1 F k,n+1 − F 2 k,n = (−1) n .

The k-Fibonacci numbers and the set
Then,  is an abelian field, with the identity element being (0, 1), and the inverse of the element (a, b) ≠ (0, 0).

From
Formula The previous definition and the results obtained allow us to find some properties of the k-Fibonacci numbers, previously proven in papers (Falcon & Plaza, 2007a, 2007b, 2009a, in the next subsection. ✷

Convolution of the k-Fibonacci numbers
It is obvious that On the other hand, and taking into account the Simson Identity, Equating first elements of this pair with (5), we deduce the convolution formula (Falcon & Plaza, 2007b;Vajda, 1989) And as particular cases, we will mention the following:

k-Fibonacci numbers of the half index
In this section, we will study the k-Fibonacci numbers of the half index.
We will call F k, 2n+1 2 a k-Fibonacci number of the half index.
Taking into account (1, 0) 1∕2 (1, 0) 1∕2 = (1, 0) and Equation (1), it is . Hence, applying definition: Then, we obtain the system of quadratic equations From the second equation, we obtain F k,−1∕2 = ± i F k,1∕2 . If we suppose the real part of the complex number F k,1∕2 is positive, then we must take F k,−1∕2 = −i F k,1∕2 . Replacing in (6), the following equation holds: Hence, we can accept for the k-Fibonacci numbers of index − 2n+1 2 the following definition: . This formula is very similar to Formula (4) for the k-Fibonacci numbers of negative integer indices.

Binnet identity
The Binnet identity for the k-Fibonacci numbers of integer indices (Falcon & Plaza, 2007b) continues being valid for the case of that n = 2r+1 2 because its characteristic equation is the same in both cases, r 2 − k r − 1 = 0. This shows that we could have defined the k-Fibonacci numbers from this formula and then to then found the different sequences for k = 1, 2, ….
It is noteworthy that many of the general formulas found for the k-Fibonacci numbers continue checking for the case of n = (2r + 1)∕2, except perhaps that sometimes it is necessary to multiply by the factor i = √ −1. We have proved in Plaza (2007a, 2007b) and in this same paper, Next, we will prove that both formulas are also valid for any number if we take into account the number of the half index. From the preceding formulas, From the second terms of both pairs, F k,n = F 2 And from the first terms, Finally, if we substitue n by 2n in these formulae, we find both initial formulas.
Also the convolution formula remains valid, and its proof is similar to the preceding, from (1, 0) n+m = (1, 0) 2n−1 2 (1, 0) 2m+1 2 and we would obtain: However, the Catalan formula for the k-Fibonacci numbers dictates that if n is an integer number (Falcon & Plaza, 2007a, 2007b, 2009a, then F k,n−r F k,n+r − F 2 k,n = (−1) n−r−1 F 2 k,r changes if the number is of the half index. In this case, the Catalan formula takes the form It is enough to apply the Binnet Identity, taking into ac-

Some notes about the k-Fibonacci numbers of the half index
(1) Both Real and imaginary parts of F k, 2n+1 2 never are integers. Consequently, F k, 2n+1 2 never is a Gaussian Integer (Weisstein, 2009).
(2) For a fixed number k, it is verified that F k,n < (3) Taking into account that | 2 | decreases when n increases, the absolute value of F k, 2n+1 2 tends to the Real part of this number:

On the sequences of k-Fibonacci numbers of half index
Let us consider the k-Fibonacci sequence of complex numbers F k, 2n+1 2 n∈ℕ . The Binnet Identity can be indicated as F k, 2n+1 Hence, the real part of the first term of this sequence is Re F k, 1 2 = √ k 2 + 4 and the real parts of the successive terms is obtained multiplying the real part of the previous term by . Similarly, the imaginary part of the first term of this sequence is Im and the imaginary parts of the successive terms are obtained by multiplying the imaginary part of the previous term by Consequently, this k-Fibonacci sequence takes the form From Equations (8) and (10), we deduce that the sequence |F k, 2n+1 2 | diverges.
From Equation (9), we obtain the following interesting results.

Theorem
For all n ∈ ℕ, following equalities hold: (1) The first two formulae are obvious. As for the third, we must bear in mind that the imaginary part tends to zero when the index tends to infinite, so its contribution to the modulus of the complex number decreases when n increases. In consequence, The next theorem relates the k-Fibonacci numbers of the half index to the k-Fibonacci numbers of the integer index.

Theorem
For all integers k and for all n ∈ ℕ: Proof Applying the Binnet identity to both sides of this equation, taking into account ( 1 2 ) 1 2 = −i, and after removing k 2 + 4 from both denominators, it becomes In particular, for n = r = 0, Equation (7)

On the sequences of k-Fibonacci numbers of the half index
Taking into acccount a k-Fibonacci number of the half index is a complex number which both realpart and imaginary part are never integers, the sequences of these numbers do not have greater interest. Of course, in these sequences, the initial relation is verified, that is F k,n+1 = k F k,n + F k,n−1 .
The sequences related to the modulus of these complex numbers are more interesting.
Let |F k,r | be the modulus of the k-Fibonacci number F k,r , when r = 2n+1 2 .
The floor function of |F k,n | is the integral part of this number: O k,n = Floor[|F k,n |]. We can also say that they are obtained by the rounding down of |F k,n |.
The round function of |F k,n | is the closest integer to this number: R k,n = Round[|F k,n |].
The ceiling function of |F k,n | is the function whose value is the smallest integer, not less than |F k,n |: C k,n = Ceiling[|F k,n |] (http://en.wikipedia.org/wiki/Catalan_number). We can also say that they are obtained by the rounding up of |F k,n |.

Expression of a k-Fibonacci number whose index is a multiple of another index
Equation (13)  In short: (1) If a ≠ 0 and b ≠ 0: (a, b) n = n ∑ j=0 n j a n−j b j (F k,n−j , F k,n−j−1 ) (2) If b = 0: (a, 0) n = a n (1, 0) n = a n (F k,n , F k,n−1 )

Integer sequences of coefficients from
(1, 1) n In the sequel, we give the expressions of the first terms of this sequence {(1, 1) n }, for n = 0, 1, 2, … With the coefficients of the first terms of the pairs of the Second-Hand Side, we form Table 1.