# Incomplete Bivariate Fibonacci and Lucasp-Polynomials

Authors: Dursun Tasci; Mirac Cetin Firengiz; Naim Tuglu
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We define the incomplete bivariate Fibonacci and Lucasp-polynomials. In the case x=1, y=1, we obtain the incomplete Fibonacci and Lucas p-numbers. If x=2, y=1, we have the incomplete Pell and Pell-Lucas p-numbers. On choosing x=1, y=2, we get the incomplete generalized Jacobsthal number and besides for p=1 the incomplete generalized Jacobsthal-Lucas numbers. In the case x=1, y=1, p=1, we have the incomplete Fibonacci and Lucas numbers. If x=1, y=1, p=1, k=⌊(n-1)/(p+1)⌋, we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas p-polynomials are given.


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## Body

## 1. Introduction

Djordjević introduced incomplete generalized Fibonacci and Lucas numbers using explicit formulas of generalized Fibonacci and Lucas numbers in [1]. In [2] incomplete Fibonacci and Lucas numbers are given as follows:
(1.1)Fn(k)=∑j=0k(n-1-jj),0≤k≤⌊n-12⌋,Ln(k)=∑j=0knn-j(n-jj),0≤k≤⌊n2⌋,
where n=1,2,3,…. Note that for the case k=⌊(n-1)/2⌋ incomplete Fibonacci numbers are reduced to Fibonacci numbers and for the case k=⌊n/2⌋ incomplete Lucas numbers are reduced to Lucas numbers in [2]. Also the authors considered the generating functions of the incomplete Fibonacci and Lucas numbers in [3]. In [4] Djordjević and Srivastava defined incomplete generalized Jacobsthal and Jacobsthal-Lucas numbers.The generalized Fibonacci and Lucasp-numbers were studied in [5, 6]. Incomplete Fibonacci and Lucas p-numbers are defined by
(1.2)Fpk(n)=∑j=0k(n-jp-1j),0≤k≤⌊n-1p+1⌋,Lpk(n)=∑j=0knn-jp(n-jpj),0≤k≤⌊np+1⌋,
for n≥1 in [7]. In [8] the authors introduced incomplete Pell and Pell-Lucas p-numbers.The generalized bivariate Fibonaccip-polynomials Fp,n(x,y) and generalized bivariate Lucas p-polynomials Lp,n(x,y) are defined the recursion for p≥1(1.3)Fp,n(x,y)=xFp,n-1(x,y)+yFp,n-p-1(x,y),n>p,
with(1.4)Fp,0(x,y)=0,Fp,n(x,y)=xn-1forn=1,2,…p,
and
(1.5)Lp,n(x,y)=xLp,n-1(x,y)+yLp,n-p-1(x,y),n>p,
with
(1.6)Lp,0(x,y)=p+1,Lp,n(x,y)=xnforn=1,2,…p
in [5]. When x=y=1, Fp,n(1,1)=Fp(n). In [5], the authors obtained some relations for these polynomials sequences. In addition, in [5], the explicit formula of bivariate Fibonacci p-polynomials is
(1.7)Fp,n(x,y)=∑j=0⌊(n-1)/(p+1)⌋(n-jp-1j)xn-j(p+1)-1yj,n≥0,p≥1,
and the explicit formula of bivariate Lucas p-polynomials is
(1.8)Lp,n(x,y)=∑j=0⌊n/(p+1)⌋nn-jp(n-jpj)xn-j(p+1)yj,n≥0,p≥1.
In this paper, we defined incomplete bivariate Fibonacci and Lucas p-polynomials. We generalize incomplete Fibonacci and Lucas numbers, incomplete generalized Fibonacci numbers, incomplete generalized Jacobsthal numbers, incomplete Fibonacci and Lucas p-numbers, incomplete Pell and Pell-Lucas p-numbers.


## 2. Incomplete Bivariate Fibonacci and Lucasp-Polynomials

Definition 2.1.
Forp≥1, n≥1, incomplete bivariate Fibonacci p-polynomials are defined as
(2.1)Fp,nk(x,y)=∑j=0k(n-jp-1j)xn-j(p+1)-1yj,0≤k≤⌊n-1p+1⌋.Forx=1, y=1, Fp,nk(x,y)=Fpk(n), we get incomplete Fibonacci p-numbers [7].Ifx=2, y=1, Fp,nk(x,y)=Ppk(n), we obtained incomplete Pell p-numbers [8].On choosingx=1, y=2, Fp,nk(x,y)=Jn,p+1k, we have incomplete generalized Jacobsthal numbers [4].Ifx=1, y=1, p=1, Fp,nk(x,y)=Fn(k), we get incomplete Fibonacci numbers [2].Forx=1,y=1,p=1,k=⌊(n-1)/(p+1)⌋,Fp,nk(x,y)=Fn, we obtained Fibonacci numbers [9].Definition 2.2.
Forp≥1, n≥1, incomplete bivariate Lucas p-polynomials are defined as
(2.2)Lp,nk(x,y)=∑j=0knn-jp(n-jpj)xn-j(p+1)yj,0≤k≤⌊np+1⌋.Ifx=1, y=1, Lp,nk(x,y)=Lpk(n), we obtained incomplete Lucas p-numbers [7].Forx=2, y=1, Lp,nk(x,y)=Qpk(n), we have incomplete Pell-Lucas p-numbers [8].On choosingx=1, y=2, p=1, Lp,nk(x,y)=jn,p+1k, we get incomplete generalized Jacobsthal-Lucas numbers [4].Ifx=1, y=1, p=1, Lp,nk(x,y)=Ln(k), we obtained incomplete Lucas numbers [2].Forx=1, y=1, p=1,k=⌊n/(p+1)⌋, Lp,nk(x,y)=Ln, we have Lucas numbers [9].Proposition 2.3.
The incomplete bivariate Fibonaccip-polynomials satisfy the following recurrence relation:
(2.3)Fp,nk+1(x,y)=xFp,n-1k+1(x,y)+yFp,n-p-1k(x,y),0≤k≤n-p-3p+1.Proof.
Using (2.1), we obtain
(2.4)xFp,n-1k+1(x,y)+yFp,n-p-1k(x,y)=x∑j=0k+1(n-jp-2j)xn-j(p+1)-2yj+y∑j=0k(n-p-pj-2j)xn-p-j(p+1)-2yj=∑j=0k+1(n-jp-2j)xn-j(p+1)-1yj+∑j=0k(n-p-pj-2j)xn-p-j(p+1)-2yj+1=∑j=0k+1(n-jp-2j)xn-j(p+1)-1yj+∑j=1k+1(n-pj-2j-1)xn-j(p+1)-1yj=∑j=0k+1[(n-jp-2j)+(n-pj-2j-1)]xn-j(p+1)-1yj-(n-2-1)xn-1=∑j=0k+1(n-jp-1j)xn-j(p+1)-1yj-0=Fp,nk+1(x,y).Takingx=y=1 in (2.3), we could obtain a formula for incomplete Fibonacci p-numbers (see [7, Proposition  3]). Taking x=y=p=1 in (2.3), we could obtain a formula for incomplete Fibonacci numbers (see [2, Proposition  1]).Proposition 2.4.
The nonhomogeneous recurrence relation of incomplete bivariate Fibonaccip-polynomials is
(2.5)Fp,nk(x,y)=xFp,n-1k(x,y)+yFp,n-p-1k(x,y)-(n-p(k+1)-2k)xn-p(k+1)-k-2yk+1.Proof.
It is easy to obtain from (2.1) and (2.3).Proposition 2.5.
For0≤k≤(n-h-p-1)/(p+1), one has
(2.6)∑j=0h(hj)yh-jxjFp,n+p(j-1)k+j(x,y)=Fp,n+(p+1)h-pk+h(x,y).Proof.
Equation (2.6) clearly holds for h=0. Suppose that the equation holds for h>0. We show that the equation holds for (h+1). We have
(2.7)∑j=0h+1(h+1j)yh+1-jxjFp,n+p(j-1)k+j(x,y)=∑j=0h+1[(hj)+(hj-1)]yh+1-jxjFp,n+p(j-1)k+j(x,y)=∑j=0h+1(hj)yh+1-jxjFp,n+p(j-1)k+j(x,y)+∑j=0h+1(hj-1)yh+1-jxjFp,n+p(j-1)k+j(x,y)=yFp,n+(p+1)h-pk+h(x,y)+(hh+1)xh+1Fp,n+phk+h+1(x,y)+∑j=-1h(hj)yh-jxj+1Fp,n+pjk+j+1(x,y)=yFp,n+(p+1)h-pk+h(x,y)+x∑j=0h(hj)yh-jxjFp,n+pjk+j+1(x,y)+(h-1)yh+1Fp,n-pk(x,y)=yFp,n+(p+1)h-pk+h(x,y)+xFp,n+(p+1)hk+h+1(x,y)=Fp,n+(p+1)h+1k+h+1(x,y).Proposition 2.6.
Forn≥k(p+1)+p+2,
(2.8)∑j=0h-1yxjFp,n-p+jk(x,y)=1xh-1Fp,n+hk+1(x,y)-xFp,nk+1(x,y).Proof.
Equation (2.8) can be easily proved by using (2.3) and induction on h.We have the following proposition in which the relationship between the incomplete bivariate Fibonacci and Lucasp-polynomials is preserved as found in [5] before.Proposition 2.7.
One has(2.9)Lp,nk(x,y)=Fp,n+1k(x,y)+pyFp,n-pk-1(x,y),0≤k≤⌊np+1⌋.Proof.
By (2.1), rewrite the right-hand side of (2.9) as
(2.10)Fp,n+1k(x,y)+pyFp,n-pk-1(x,y)=∑j=0k(n-jpj)xn-j(p+1)yj+py∑j=0k-1(n-p-jp-1j)xn-p-j(p+1)-1yj=∑j=0k(n-jpj)xn-j(p+1)yj+py∑j=1k(n-jp-1j-1)xn-j(p+1)yj-1=∑j=0k[(n-jpj)+(n-jp-1j-1)]xn-j(p+1)yj-(n-1-1)xn=∑j=0knn-jp(n-jpj)xn-j(p+1)yj=Lp,nk(x,y).Proposition 2.8.
The incomplete bivariate Lucasp-polynomials satisfy the following recurrence relation:
(2.11)Lp,nk+1(x,y)=xLp,n-1k+1(x,y)+yLp,n-p-1k(x,y),0≤k≤n-p-2p+1.Proof.
We write by using (2.3) and (2.9)
(2.12)Lp,nk+1(x,y)=Fp,n+1k+1(x,y)+pyFp,n-pk(x,y)=xFp,nk+1(x,y)+yFp,n-pk(x,y)+py[xFp,n-p-1k(x,y)+yFp,n-2p-1k-1(x,y)]=x[Fp,nk+1(x,y)+pyFp,n-p-1k(x,y)]+y[Fp,n-pk(x,y)+pyFp,n-2p-1k-1(x,y)]=xLp,n-1k+1(x,y)+yLp,n-p-1k(x,y).Proposition 2.9.
The nonhomogeneous recurrence relation of incomplete bivariate Lucasp-polynomials is
(2.13)Lp,nk(x,y)=xLp,n-1k(x,y)+yLp,n-p-1k(x,y)-n-p-1n-p(k+1)-1(n-p(k+1)-1k)xn-(p+1)(k+1)yk+1.Proof.
The proof can be done by using (2.2) and (2.11).Proposition 2.10.
For0≤k≤(n-p-h)/(p+1), one has
(2.14)∑j=0h(hj)xjyh-jLp,n+p(j-1)k+j(x,y)=Lp,n+(p+1)h-pk+h(x,y).Proof.
Proof is similar to the proof of Proposition2.5.Proposition 2.11.
Forn≥(k+1)(p+1), one has
(2.15)∑j=0h-1yxjLp,n-p+jk(x,y)=1xh-1Lp,n+hk+1(x,y)-xLp,nk+1(x,y).Proof.
Proof is obtained immediately by using (2.11) and induction h.Proposition 2.12.
One has(2.16)∑k=0⌊n/(p+1)⌋Lp,nk(x,y)=(⌊np+1⌋+1)Lp,n(x,y)+np+1[xFp,n(x,y)-Lp,n(x,y)].Proof.
We can write from (2.2)
(2.17)∑k=0⌊n/(p+1)⌋Lp,nk(x,y)=Lp,n0(x,y)+Lp,n1(x,y)+Lp,n2(x,y)+⋯+Lp,n⌊n/(p+1)⌋(x,y)=nn(n0)xn+[nn(n0)xn+nn-p(n-p1)xn-(p+1)y]+[nn(n0)xn+nn-p(n-p1)xn-(p+1)y+nn-2p(n-2p2)xn-2(p+1)y2]+⋯+[(n-⌊np+1⌋p⌊np+1⌋)xn-(p+1)⌊np+1⌋y⌊np+1⌋nn(n0)xn+⋯+nn-⌊n/(p+1)⌋p×(n-⌊np+1⌋p⌊np+1⌋)xn-(p+1)⌊n/(p+1)⌋y⌊n/(p+1)⌋]=(⌊np+1⌋+1)nn(n0)xn+(⌊np+1⌋+1-1)nn-p(n-p1)xn-(p+1)y+(⌊np+1⌋+1-2)nn-2p(n-2p1)xn-2(p+1)y2+⋯+nn-⌊n/(p+1)⌋p(n-⌊np+1⌋p⌊np+1⌋)xn-(p+1)⌊n/(p+1)⌋y⌊n/(p+1)⌋=∑j=0⌊n/(p+1)⌋(⌊np+1⌋+1-j)nn-jp(n-jpj)xn-j(p+1)yj=(⌊np+1⌋+1)∑j=0⌊n/(p+1)⌋nn-jp(n-jpj)xn-j(p+1)yj-∑j=0⌊n/(p+1)⌋jnn-jp(n-jpj)xn-j(p+1)yj.
Equation (2.17) is calculated using the formula Lp,n(x,y) and ∂Lp,n(x,y)/∂x=nFp,n(x,y) [5]
(2.18)∑k=0⌊n/(p+1)⌋Lp,nk(x,y)=(⌊np+1⌋+1)∑j=0⌊n/(p+1)⌋nn-jp(n-jpj)xn-j(p+1)yj+nFp,n(x,y)-nx-1Lp,n(x,y)(p+1)x-1=(⌊np+1⌋+1)Lp,n(x,y)+np+1[xFp,n(x,y)-Lp,n(x,y)].Then we have the following conclusion.Conclusion.
Whenx=y=p=1 in (2.16), we obtain
(2.19)∑k=0⌊n/2⌋Ln(k)=(⌊n2⌋+1)Ln+n2(Fn-Ln)
which is Proposition  11 in [2].


## 3. Generating Functions of the Incomplete Bivariate Fibonacci and Lucasp-Polynomials

Lemma 3.1 (see [3]).
Let{sn}n=0∞ be a complex sequence satisfying the following nonhomogeneous recurrence relation:
(3.1)sn=xsn-1+ysn-p-1+rn,n>p,
where {rn} is a given complex sequence. Then the generating function Spk(x,y;t) of the sequence {sn} is
(3.2)Spk(x,y;t)=[s0-r0+∑i=1p(si-xsi-1-ri)ti+G(t)][1-xt-ytp+1]-1,
where G(t) denotes the generating function of {rn}.Theorem 3.2.
The generating function of the incomplete bivariate Fibonaccip-polynomials is
(3.3)Rpk(x,y;t)=tk(p+1)+1[Fp,k(p+1)+1(x,y)+∑i=1pti(Fp,k(p+1)+1+i(x,y)-xFp,k(p+1)+i(x,y))+yk+1tp+1(1-xt)k+1][1-xt-ytp+1]-1.Proof.
From (2.1) and (2.5), Fp,nk(x,y)=0for0≤n<k(p+1)+1,
(3.4)Fp,k(p+1)+1k(x,y)=Fp,k(p+1)+1(x,y),Fp,k(p+1)+2k(x,y)=Fp,k(p+1)+2(x,y),⋮Fp,k(p+1)+p+1k(x,y)=Fp,k(p+1)+p+1(x,y),
and for n≥k(p+1)+p+2
(3.5)Fp,nk(x,y)=xFp,n-1k(x,y)+yFp,n-p-1k(x,y)-(n-p(k+1)-2n-k(p+1)-p-2)xn-p(k+1)-k-2yk+1.
Now let
(3.6)s0=Fp,k(p+1)+1k(x,y),s1=Fp,k(p+1)+2k(x,y),…,sp=Fp,k(p+1)+p+1k(x,y),
and
(3.7)sn=Fp,n+k(p+1)+1k(x,y).
Also
(3.8)r0=r1=⋯=rp=0,rn=(n+k-p-1n-p-1)xn-p-1yk+1.
We obtained that G(t)=yk+1tp+1/(1-xt)k+1 is the generating function of the sequence {rn}. From Lemma  3.1, we get that the generating function Spk(x,y;t) of sequence {sn} is
(3.9)Spk(x,y;t)=[Fp,k(p+1)+1k(x,y)+∑i=1pti(Fp,k(p+1)+1+ik(x,y)-xFp,k(p+1)+ik(x,y))+yk+1tp+1(1-xt)k+1][1-xt-ytp+1]-1.
Therefore,
(3.10)Rpk(x,y;t)=tk(p+1)+1Spk(x,y;t).Theorem 3.3.
The generating function of the incomplete bivariate Lucasp-polynomials is
(3.11)Wpk(x,y;t)=tk(p+1)[Lp,k(p+1)(x,y)+∑i=1pti(Lp,k(p+1)+i(x,y)-xLp,k(p+1)+i-1(x,y))+tp+1yk+1[p(1-xt)+1](1-xt)k+1][1-xt-ytp+1]-1.Proof.
From (2.9) and (3.3),
(3.12)Wpk(x,y;t)=∑n=0∞Lp,nk(x,y)tn=∑n=0∞[Fp,n+1k(x,y)+pyFp,n-pk-1(x,y)]tn=∑n=0∞Fp,n+1k(x,y)tn+py∑n=0∞Fp,n-pk-1(x,y)tn=t-1Rpk(x,y;t)+pytpRpk-1(x,y;t).
For the general case in Theorems3.2 and 3.3, we find the generating functions of some special numbers by the special cases x,y,p. For example, x=y=1 in (3.3) we obtain the generating function of incomplete Fibonacci p-numbers.



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