A Note on Incomplete Fibonacci–Lucas Relations

: We deﬁne the incomplete generalized bivariate Fibonacci p -polynomials and the incomplete generalized bivariate Lucas p -polynomials. We study their recursive relations and derive an interesting relationship through their generating functions. Subsequently, we prove an incomplete version of the well-known Fibonacci–Lucas relation and make some extensions to the relation involving incomplete generalized bivariate Fibonacci and Lucas p -polynomials. An argument about going from the regular to the incomplete Fibonacci–Lucas relation is discussed. We provide a relation involving the incomplete Leonardo and the incomplete Lucas–Leonardo p -numbers as an illustration.


Introduction
In [1], Filipponi investigated and obtained many properties of the incomplete Fibonacci numbers and the incomplete Lucas numbers.For a real number x, x denotes the least integer greater than or equal to x.For any positive integer n, the incomplete Fibonacci numbers F n (s) are defined as where s is an integer with 0 ≤ s ≤ n−1

2
. Similarly, the incomplete Lucas numbers L n (s) are defined by where 0 ≤ s ≤ n 2 .Note that F n ( n−1

2
) is equal to the original Fibonacci number F n , and L n ( n 2 ) = L n is the Lucas number, and this is a part of the reason that the name "incomplete" is used.Additionally, for our convenience, let L 0 (0) = 2.Some special cases of ( 1) and ( 2) are F n (0) = L n (0) = 1, for all n ≥ 1, and F n (1) = n − 1, for all n ≥ 3; L n (1) = n + 1, for all n ≥ 2.
Sury [2] proved the well-known Fibonacci-Lucas relation: Chung [3] proved a more general relation for the sequence of W-polynomials and wpolynomials.Chung, Yao, and Zhou [4] extended Sury's formula (3) in both a regular and an alternating form to Fibonacci k-step and Lucas k-step polynomials.
Equivalently, we rewrite (3) as ) = F n and L n ( n 2 ) = L n .Now we can extend the Fibonacci-Lucas relation to the incomplete version: for all positive integer n and 0 ≤ s ≤ n 2 .To see (4), it is suffice to show that, for all s with 0 ≤ s ≤ n 2 , By the well-known Fibonacci-Lucas relation (3), we have 2s Identity (5) can be proved by induction on n, or it can be proved directly as follows.See also (3.13) in [1].
Lemma 1.For any positive integer n, we have Proof.By the definitions of ( 1) and (2), for s with 0 ≤ s ≤ n−1

2
, we have Now, from the previous lemma, we know that the right-hand side of ( 5) is equal to If n is even and s = n 2 , (5) holds obviously.Thus (5) or the incomplete Fibonacci-Lucas relation (4) follows.
In 2012, Tasci, Cetin Firengiz, and Tuglu [5] investigated the incomplete bivariate Fibonacci and Lucas p-polynomials: where 0 ≤ s ≤ n−1 p+1 , and n;p (x, y) reduces to the bivariate Fibonacci p-polynomials F n;p (x, y).That is, In [5], some basic properties and generating functions of the incomplete bivariate Fibonacci and Lucas p-polynomials are given.In this note, we define the incomplete generalized bivariate Fibonacci and Lucas p-polynomials as below.For any two integers p ≥ 1, n ≥ 0 and any two polynomials h(x, y), (x, y) with real coefficients, define where 0 ≤ s ≤ n−1 p+1 , and n;p (h, ) if there is no misunderstanding.We are at the stage of stating the following main theorem.Theorem 1.For any integers p ≥ 1 and n ≥ 0, any real nonzero number r, and any polynomials h(x, y), (x, y) ∈ R[x, y], we have a relation involving the incomplete generalized bivariate Fibonacci and Lucas p-polynomials, where s is any integer with 0 ≤ s ≤ n p+1 .
We replace r in Theorem 1 with −1/r (since r = 0) to obtain an alternating relation involving the incomplete generalized bivariate Fibonacci and Lucas p-polynomials: In the case of h(x, y) = x and (x, y) = y in Theorem 1, we have a relation and an alternating relation involving the incomplete bivariate Fibonacci and Lucas p-polynomials, respectively.Corollary 1.For any integers p ≥ 1 and n ≥ 0 and any real nonzero number r, we have where s is any integer with 0 ≤ s ≤ n p+1 .
In the case of h(x, y) being just a polynomial of x, say h(x), and (x, y) = 1 = p in Theorem 1, we obtain a relation involving the incomplete h(x)-Fibonacci F (s) Corollary 2. For any integer n ≥ 0, any real nonzero number r, and a polynomial h(x) with a real coefficient, we have where s is any integer with 0 ≤ s ≤ n 2 .
In the case of h(x, y) = (x, y) = 1 and p = 1 in Theorem 1, we have the following generalized incomplete Fibonacci-Lucas relation.

Corollary 3.
For any integer n ≥ 0 and any real nonzero number r, we have where s is any integer with 0 ≤ s ≤ n 2 .
This note is organized as follows.In Section 2, we establish an inter-relationship between the incomplete generalized bivariate Fibonacci p-polynomials and the incomplete generalized bivariate Lucas p-polynomials and investigate some properties of these polynomials.Afterwards, we derive both of the two generating functions, and from these, we can obtain an interesting relationship between the two generating functions (Proposition 6).We then give proof of our main theorem (Theorem 1).In Section 3, we discuss the regular generalized bivariate Fibonacci and Lucas p-polynomials and obtain a potential connection between the regular (complete) and incomplete Fibonacci-Lucas relation.We also discuss, as an example, a relation involving the Leonardo p-numbers and the Lucas-Leonardo pnumbers.We show a procedure for how to obtain such a relation in an incomplete version from a regular (complete) form.A summary and conclusion will be given in Section 4.

Some Properties and Proofs
In this section, let p be a positive integer and n ≥ 0 be an integer.We note that, from the definitions of the incomplete generalized bivariate Fibonacci and Lucas p-polynomials, and Proposition 1.The incomplete generalized bivariate Fibonacci p-polynomials satisfy a nonhomogeneous recurrence relation: for all n ≥ p + 2 and 0 ≤ s ≤ n−p−2 p+1 .
Proof.For n ≥ p + 2 and 0 ≤ s ≤ n−p−2 p+1 , we have It is easy to see that the recurrence relation ( 6) can be written in a homogeneous form: Proposition 2. For all integer t ≥ 0, and 0 n+(p+1)t−p;p (h, ).
Proof.For the case t = 0, the identity holds trivially.Assume that the desired identity holds for some t > 0. Now, for 0 Thus, by induction on t, the desired identity follows for all t ≥ 0.
Similarly, we obtain the recurrence relation for the incomplete generalized bivariate Lucas p-polynomials, where n ≥ p + 1 and 0 ≤ s ≤ n−p−1 p+1 .Equivalently, a nonhomogeneous recursion is given by By a similar argument to the proof of Proposition 2, we obtain the following result.

n+(p+1)t−p;p (h, ).
There is an identity between the incomplete generalized bivariate Fibonacci p-polynomials and the incomplete generalized bivariate Lucas p-polynomials.
Proof.It can be deduced directly from the definition.
We define the generating function of the incomplete generalized bivariate Fibonacci n;p (h, ) = 0 for n < s(p + 1) + 1, we see that s(p+1)+1+j;p (h, )z j and then R Proof.We write In light of (6), the above right-hand side is equal to Let the generating function of the incomplete Fibonacci numbers F n (s) be According to Proposition 4, the special case p = 1 and h(x, y) = (x, y) = 1 gives Because F 2s+1 (s) is the Fibonacci number F 2s+1 and also F 2s+2 (s) = F 2s+2 , we obtain the following corollary.
Corollary 4. Let R s (z) be the generating function of the incomplete Fibonacci numbers F n (s).We have We now define the generating function of the incomplete generalized bivariate Lucas n;p (h, )z n .By (7), we have From this, we further have the following result.

Proposition 5. The generating function T (s)
p (h, ; z) of the incomplete generalized bivariate Lucas p-polynomials is given by Corollary 5. Let T s (z) be the generating function of the incomplete Lucas numbers L n (s).We have From the previous two propositions, we cancel all inhomogeneous terms of the two representations of the generating function and consider After careful calculation, we obtain the following result.Proposition 6. Notations as above, for all s ≥ 1, we have zT We remark here.We use only relation (6) (see Proposition 1) when proving Proposition 6, and do not use relation (7) in Lemma 2. To see (7) for another proof, we compare the coefficient z n+1 on both sides of Equation ( 8) to obtain where 0 ≤ s ≤ n−1 p+1 .Replacing s in (9) with s + 1 and using the recurrence relation of F (s+1) n;p (h, ), we obtain relation (7): In the very special case p = 1 and h(x, y) = (x, y) = 1 of (8), we obtain Comparing the coefficients of z n+1 on both sides of the above equation, we obtain We are now in a position to prove our main theorem.
In light of (9), the right-hand side is equal to If n is a multiplier of p + 1 and s = n p+1 , the desired relation holds obviously.This proves Theorem 1.

From Complete to Incomplete
Actually, one may start with the regular Fibonacci-Lucas relation.Given two polynomials h(x, y), (x, y) ∈ R[x, y], and an integer p ≥ 1.Let the generalized bivariate Fibonacci and Lucas p-polynomials be defined by recursive relations: with initial conditions with initial conditions It is not difficult to derive the explicit formulas of these polynomials.
Proposition 7. The explicit formula of generalized bivariate Fibonacci p-polynomials F n;p (h, ) is and the explicit formula of generalized bivariate Lucas p-polynomials L n;p (h, ) is given by must be equal to ph(x, y)r s(p+1)+1 F (s) for s is any integer with 0 ≤ s ≤ n p+1 .From this and using Equation (9) to make a telescopic sum, one can deduce the incomplete version of a generalized Fibonacci-Lucas relation, and obtain a relation involving the incomplete generalized bivariate Fibonacci and Lucas p-polynomial as Theorem 1.
Here is another example.For any integer p ≥ 1, let the Leonardo p-numbers L n;p be defined by the following nonhomogeneous recurrence relation: for n ≥ p + 1 with initial conditions L 0;p = L 1;p = • • • = L p;p = 1.Tan and Leung [7] introduced the Leonardo p-sequence {L n;p } n≥0 as a generalization of classical Leonardo numbers.The Leonardo 1-numbers, simply denoted by L n , are the classical Leonardo numbers that represent the number of vertices in the n-th Leonardo tree.That is, L n satisfies the relation: One can show easily that all classical Leonardo numbers are odd.Tan and Leung [7] investigated some basic properties of Leonardo p-numbers and derived some relations between the Leonardo p-numbers and the Fibonacci p-numbers (by letting h(x, y) = (x, y) = 1 in (10)), such as and in particular L n = 2F n+1 − 1.
Recently, Tan and Leung [7] investigated incomplete Leonardo p-numbers and gave some properties of these numbers.Indeed, in their paper [7], they defined the incomplete Leonardo p-numbers as where s is an integer with 0 ≤ s ≤ n p+1 .From this definition and (15), it is clear to see that L n;p (0) = 1, L n;p (1) = (p + 1)(n − p) + 1, and L n;p ( n p+1 ) = L n;p .The incomplete Leonardo p-numbers L n;p (s) satisfy the recurrence relation . One can find proof in [7].Similarly, we consider relation ( 16) and may define the incomplete Lucas-Leonardo p-numbers as below.For integers n ≥ 0 and p ≥ 1 and an integer s with 0 ≤ s ≤ n p+1 , we define Some special cases of the above definition are 1.
K n;p ( n p+1 ) = (p + 1)L n;p − p = K n;p .Furthermore, we have the following proposition.It is easy to see that relation (18) can be transformed into the nonhomogeneous recurrence relation: for 0 ≤ s ≤ n−1 p+1 .

Proof.
K