A new hybrid generalization of Fibonacci

. The hybrid numbers are generalization of complex, hyperbolic and dual numbers. The hybrinomials are polynomials which generalize hybrid numbers. In this paper, we introduce and study the distance Fibonacci hy-brinomials, i.e. hybrinomials with coefficients being distance Fibonacci polynomials.


Introduction.
The hybrid numbers were introduced by Özdemir in [13] as a new generalization of complex, hyperbolic and dual numbers.
Let K be the set of hybrid numbers Z of the form where the coefficients a, b, c, d are real numbers and i, ε, h are operators such that (1) The addition and subtraction of hybrid numbers are done by adding and subtracting corresponding terms and hence their coefficients.The hybrid numbers multiplication is defined using (1) and (2).Note that using the formulas (1) and (2), we can find the product of any two hybrid units.The following Table 1 presents products of i, ε, and h.
Table 1.The hybrid numbers multiplication.
Using the rules given in Table 1, the multiplication of hybrid numbers can be made analogously as multiplications of algebraic expressions.Moreover, (K, +, •) is a non-commutative ring.
The Fibonacci numbers F n are defined recursively by F n = F n−1 + F n−2 for n ≥ 2 with initial terms F 0 = 0, F 1 = 1.The Lucas numbers L n are defined by L n = L n−1 + L n−2 for n ≥ 2 with L 0 = 2, L 1 = 1.The Fibonacci-Narayana numbers N n are defined as follows N n = N n−1 + N n−3 for n ≥ 3 with N 0 = 0, N 1 = 1, N 2 = 1, for details see [10].
For any variable quantity x, the Fibonacci polynomials F n (x) are defined as For x = 1 the Fibonacci, Lucas and Fibonacci-Narayana polynomials give the Fibonacci, Lucas and Fibonacci-Narayana numbers, respectively.Properties of Fibonacci, Lucas and Fibonacci-Narayana polynomials can be found in [5,6,7,9,14,17,23,24], among others.In recent years, many interesting papers investigating the properties of Narayana numbers and Narayana polynomials have been published, see e.g.[8,12,15,16,18].
Fibonacci hybrid numbers were defined and studied in [19].In [20], the authors presented some properties of Fibonacci and Lucas hybrid numbers.Fibonacci-Narayana hybrid numbers (with initial conditions 1, 1, 1) were examined in [22].
The nth Fibonacci hybrid number F H n , the nth Lucas hybrid number LH n and the nth Fibonacci-Narayana hybrid number are defined as respectively.
The hybrinomials are polynomials, which are a generalization of hybrid numbers.The term "hybrinomials" was used for the first time in [21], where Fibonacci and Lucas hybrinomials were studied.The Narayana polynomials (with initial conditions 2, 3, 4) and Narayana hybrinomials were considered in [17].Some generalization of Fibonacci and Lucas hybrinomials was introduced in [1].The authors defined a class of hybrid polynomials (hybrinomials), which are so-called "r-Fibonacci hybrid polynomials and r-Lucas hybrid polynomials of type s".
We recall that for n ≥ 0 the Fibonacci and Lucas hybrinomials are defined by ( 6) where F n (x) is the nth Fibonacci polynomial, L n (x) is the the nth Lucas polynomial and i, ε, h are hybrid units which satisfy (1) and (2).By analogy, for n ≥ 0 the Fibonacci-Narayana hybrinomials are defined by ( 8) Using formulas ( 6)- (8), for x = 1 we obtain the Fibonacci hybrid numbers, the Lucas hybrid numbers and the Fibonacci-Narayana hybrid numbers, respectively.
In the literature we can find many generalizations of Fibonacci and Lucas numbers, see for example the list in [2].The authors generalized the definition of the Fibonacci numbers by changing the initial conditions, changing the recurrence relation or changing distance between terms of a sequence.One of the generalizations in the distance sense was introduced in [11] as follows.
Let k ≥ 2, n ≥ 0 be integers.The generalized Fibonacci numbers F (k, n) and generalized Lucas numbers L(k, n) were defined as Note that for n ≥ 0 we have In [22], the authors defined For k = 2 we obtain F H 2 n = F H n+2 and LH 2 n = LH n .For k = 3 we have F H 3 n = N H n+3 .In [3], the authors introduced distance Fibonacci polynomials as a generalization of Fibonacci and Fibonacci-Narayana polynomials.Let k ≥ 2, n ≥ 0 be integers.The distance Fibonacci polynomials f n (k, x) are given by the following recurrence relation Table 3 presents some distance Fibonacci polynomials f n (k, x) for special values of k and n.
Note that The generalization of Fibonacci numbers and Fibonacci polynomials is the motivation to generalize Fibonacci hybrinomials in terms of distance.Based on the definition of distance Fibonacci polynomials we will define distance Fibonacci hybrinomials in the following way.
For n ≥ 0 the distance Fibonacci hybrinomials are defined by ( 12) where f n (k, x) is the nth distance Fibonacci polynomial and i, ε, h are hybrid units which satisfy (1) and ( 2).
In the next section we will present some properties of these hybrinomials.

Main results.
We start with the recurrence relations for distance Fibonacci hybrinomials.
Theorem 1.Let k ≥ 2, n ≥ 0 be integers.For any variable quantity x, we have Proof.For an integer n, n ≥ k, using the definition of the distance Fibonacci polynomials, we have which ends the proof.□ Theorem 2. Let n ≥ 0, k ≥ 2 be integers.The generating function of the distance Fibonacci hybrinomials sequence {f H k n (x)} has the following form Proof.Assume that the generating function of the distance Fibonacci hybrinomials sequence {f Multiplying the above equality on both sides by −xt and then by −t k , we obtain Adding the above three equalities, we get for n ≥ k and the coefficients of t n for n ≥ k are equal to zero.□ As a special case, we obtain the generating function of the Fibonacci hybrinomials, given in [21].

Corollary 3. Let n ≥ 0 be an integer. The generating function of the Fibonacci hybrinomials sequence {F H n (x)} has the following form
Proof.For k = 2 we have Moreover, where G(t) denotes the generating function of the Fibonacci hybrinomials sequence {F H n (x)}.Then we have and after calculations the result follows.□ In the same way we can prove the next result.

Corollary 4.
Let n ≥ 0 be an integer.The generating function of the Fibonacci-Narayana hybrinomials sequence {N H n (x)} has the following form In [3], many properties of distance Fibonacci polynomials were given.We will recall two of them which will be useful in the next theorems.
Theorem 6 ([3]).Let k ≥ 2, n ≥ 0 be integers.Then Proof.For integers k ≥ 2, n ≥ 0 we have By Theorem 5 we get which ends the proof.□ Corollary 8. Let n ≥ 0 be an integer.Then Corollary 9. Let n ≥ 1 be an integer.Then Proof.By Theorem 6 we get which ends the proof.□ Concluding Remarks.The recurrences defining the generalized Fibonacci numbers F (k, n) and the generalized Lucas numbers L(k, n) are of the kth order.Hence, it is difficult to obtain the Binet formula for these sequences for any integer k.Applying some graph interpretation of distance Fibonacci polynomials f n (k, x), the authors of [3] derived the direct formula for f n (k, x).For integers k ≥ 2, n ≥ 0, x ≥ 1 the explicit closed form expression for the distance Fibonacci polynomial is given by the following formula Using this formula, one can give the direct formula for the nth distance Fibonacci hybrinomial, but, importantly, only for integer x.It would be useful to find the Binet formula for any x.Then some new identities, namely Catalan's, Cassini's, d'Ocagne's and Vajda's for the distance Fibonacci hybrinomials could be found.
In [4], the authors gave a new generalization of Lucas polynomials in the distance sense.Moreover, L(k, n)-Lucas hybrid numbers LH k n were investigated in [22].Our results obtained for distance Fibonacci hybrinomials may be a contribution to considerations about properties of distance Lucas hybrinomials.

Table 2
presents initial words of generalized Fibonacci numbers and generalized Lucas numbers for special cases of n and k.

Table 2 .
The values of F (k, n), L(k, n), Fn, Nn and Ln.