On ( k , p ) -Fibonacci Numbers

: In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Fibonacci numbers, Pell numbers, and Narayana numbers, simultane-ously. We prove some identities which generalize well-known relations for Fibonacci numbers, Pell numbers and their generalizations. A matrix representation for generalized Fibonacci numbers is given, too.


Introduction
By numbers of the Fibonacci type we mean numbers defined recursively by the r-th order linear recurrence relation of the form a n = b 1 a n−1 + b 2 a n−2 + · · · + b r a n−r , for n r, where r 2 and b i 0, i = 1, 2, · · · , r are integers.

Generalization and Identities
In this section we introduce (k, p)-Fibonacci numbers, denoted by F k,p (n). We prove some identities for F k,p (n), which generalize well-known relations for the Fibonacci numbers, Pell numbers, Narayana numbers, k-Fibonacci numbers, Fibonacci s-numbers and generalized Fibonacci numbers, simultaneously.
Let k 2, n 0 be integers and let p 1 be a rational number. The (k, p)-Fibonacci numbers, denoted by F k,p (n), are defined recursively in the following way with initial conditions F k,p (0) = 0 and F k,p (n) = p n−1 for 1 n k − 1.
For special values k, p we obtain well-known numbers of the Fibonacci type. We list these special cases.
We give the generating functions for the (k, p)-Fibonacci sequence. Let k 2 be integers and let p 1 be a rational number. Let us consider (k, p)-Fibonacci sequence {F k,p (n)}.
By the definition of an ordinary generating function of some sequence, considering this sequence, the ordinary generating function associated is defined by Using the initial conditions for F k,p (n) and the recurrence (2) we can write (4) as follows Consider the right side of the Equation (5) and doing some simple calculations, we obtain the following theorem.
Theorem 1. Let k 2 be integers and let p 1 be a rational number. The generating function of the sequence F k,p (n) has the following form From Theorem 1, for special values of k and p, we obtain well-known generating functions for Fibonacci numbers, Pell numbers, and and k-Fibonacci numbers.
Proof. Let k 2, n k − 2 be integers and p 1 be a rational number.
Let us consider the following cases: Then Using the initial conditions for (k, p)-Fibonacci numbers, we obtain By simple calculation Using the initial conditions for F k,p (n) and proving analogously as in case 1, we have Then (2p − 1)S = p k−1 + p − 1 + p p + p 2 + · · · + p k−2 and for n = k − 1 3. n k. Using the recurrence (2), we have and consequently Using the recurrence F k,p (n) = pF k,p (n − 1) + (p − 1)F k,p (n − k + 1) + F k,p (n − k) and the initial conditions of F k,p (n), we have what completes the proof.
For special values k and p, we obtain well-known identities.
Corollary 2. Let k 2, n k − 2, t 2 be integers and p 1 be a rational number. If Theorem 3. Let k 2, m 1, n be integers and p 1 be a rational number.
Let S 2n = n ∑ i=0 F k,p (2i). If k = 2m, then for n m − 1 If k = 2m + 1, then for n m S 2n = 1 3(2p − 1) 3F 2m+1,p (2n + 1) + (2p − 1)F 2m+1,p (2n) (9) Proof. (by induction on n). Let k 2, m 1, n be integers and p 1 be a rational number. First, we will show that for k = 2m, n m − 1 we have If n = m − 1, then using initial conditions of F k,p (n), we have so the Equality (8) is true for n = m − 1. Assume now that for an integer n m − 1 holds We shall show that Using the induction hypothesis, we obtain so, the Equality (8) is true. Similarly, we can show that for k = 2m + 1, n m: s is odd.
Using Theorem 4 and the equality n ∑ i=1 F s (i) = F s (n + s + 1) − 1 we obtain a sum of the first n − 1 even terms of F s (n).
Corollary 5. Let s 1, n 1 be integers. Then F k,p (2i + 1) and Theorems 2 and 3 we obtain the next identity for F k,p (n).
Theorem 4. Let k 2, m 1, n be integers and p 1 be a rational number. Let If k = 2m + 1, then for n m Corollary 6. Let k 2, n k−2 2 , t 2 be integers and p 1 be a rational number. If For more identities of the (k, p)-Fibonacci numbers see [12].

Matrix Generator of (k, p)-Fibonacci Numbers
In the last few decades, miscellaneous affinities between matrices and linear recurrences were studied, see, for instance [21,25]. The main aim is to obtain numbers defined by recurrences of matrices which are called generating matrices.
For the classical Fibonacci numbers, the matrix generator has the following form Q = 1 1 1 0 and it is well-known that for n 2 we have Q n = F n F n−1 F n−1 F n−2 , (see, for example, [21]). This generator gives the well known Cassini formula for the Fibonacci numbers, namely detQ n = (−1) n = F n F n−2 − F 2 n−1 .
For Pell numbers, the matrix generator has the form M = 2 1 1 0 and it is easly established that M n = P n+1 P n P n P n−1 , (see, for example, [25]).
In [8] the matrix generator for distance Fibonacci numbers was introduced. Using this idea we introduce the matrix generator for (k, p)-Fibonacci numbers, which generalizes the matrix generator for Fibonacci numbers and Pell numbers, simultaneously.
Let Q k = [q ij ] k×k . For a fixed 1 i k an element q i1 is equal to the coefficient of F k,p (n − i) in the Equality (2). Moreover for j 2 we have For k = 2, 3, 4 we obtain matrices Thus, for k > 2 we have If k = 2 and p = 1, then Q 2 is the matrix generator for Fibonacci numbers. If k = 2 and p = 3 2 , then Q 2 is the matrix generator for Pell numbers. The matrix Q k will be named as the companion matrix of the (k, p)-Fibonacci numbers or the (k, p)-Fibonacci matrix. Let A k be the matrix of initial conditions. Then Theorem 5. Let k 2, n 1 be integers and p 1 be a rational number. Then Proof. (by induction on n). Let k, n, p be as in the statement of the theorem. If n = 1 then Assume now that the formula is true for all integers 1, 2, · · · , n. We shall show that Since A k Q n+1 k = A k Q n k Q k , so by induction hypothesis and from the recurrence Formula (2) we obtain that A k Q n+1 k is equal to . . . . . . . . . . . .
Theorem 6. Let k 2, n 1 be integers. Then for an arbitrary rational p 1 holds Proof. Let k 2 be an integer. We prove only (11). Using the recurrence (2) and the initial conditions for (k, p)-Fibonacci numbers, we obtain   , which ends the proof.

Conclusions
In this paper we studied (k, p)-Fibonacci numbers which generalize, among others, Fibonacci numbers, Pell numbers and Narayana numbers. We presented properties of this numbers, including their generating function and matrix representation. It is interesting that the results obtained for the (k, p)-Fibonacci numbers generalize, among others, the results presented in  Stakhov (1977).
Based on the suggestion of the reviewer, it seems to be interesting to open a new direction of research by the assumption that the parameter p in the Equality (2) is a real number. Then some interesting results related to the characteristic equation of the sequence recurrence relations can be studied and the explicit form of these numbers perhaps will be obtained.