Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 840345, 11 pages
doi:10.1155/2012/840345
Review Article
Incomplete Bivariate Fibonacci and Lucas
p-Polynomials
Dursun Tasci,1 Mirac Cetin Firengiz,2 and Naim Tuglu1
1
2
Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, 06500 Ankara, Turkey
Department of Mathematics, Faculty of Education, Başkent University, Baglica, 06810 Ankara, Turkey
Correspondence should be addressed to Mirac Cetin Firengiz, mcetin@baskent.edu.tr
Received 26 November 2011; Accepted 8 February 2012
Academic Editor: Gerald Teschl
Copyright q 2012 Dursun Tasci et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We define the incomplete bivariate Fibonacci and Lucas p-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.
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:
Fn k k
n−1−j
j0
n−1
0≤k≤
,
2
,
j
k
n
Ln k n
−j
j0
n−j
j
,
0≤k≤
n
2
1.1
,
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
2
Discrete Dynamics in Nature and Society
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 Lucas p-numbers were studied in 5, 6. Incomplete
Fibonacci and Lucas p-numbers are defined by
Fpk n
k
n − jp − 1
k
n
Lkp n n − jp
j0
n−1
0≤k≤
,
p1
,
j
j0
n − jp
,
n
0≤k≤
,
p1
j
1.2
for n ≥ 1 in 7. In 8 the authors introduced incomplete Pell and Pell-Lucas p-numbers.
The generalized bivariate Fibonacci p-polynomials Fp,n x, y and generalized bivariate
Lucas p-polynomials Lp,n x, y are defined the recursion for p ≥ 1
Fp,n x, y xFp,n−1 x, y yFp,n−p−1 x, y ,
n > p,
1.3
with
Fp,0 x, y 0,
Fp,n x, y xn−1
for n 1, 2, . . . p,
1.4
and
Lp,n x, y xLp,n−1 x, y yLp,n−p−1 x, y ,
n > p,
1.5
with
Lp,0 x, y p 1,
Lp,n x, y xn
for n 1, 2, . . . p
1.6
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
Fp,n x, y n−1/p1
n − jp − 1
j0
j
xn−jp1−1 yj ,
n ≥ 0, p ≥ 1,
1.7
and the explicit formula of bivariate Lucas p-polynomials is
Lp,n x, y n/p1
j0
n
n − jp
n − jp
j
xn−jp1 yj ,
n ≥ 0, p ≥ 1.
1.8
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3
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 Lucas p-Polynomials
Definition 2.1. For p ≥ 1, n ≥ 1, incomplete bivariate Fibonacci p-polynomials are defined as
k
Fp,n
x, y k
n − jp − 1
j0
xn−jp1−1 yj ,
j
0≤k≤
n−1
.
p1
2.1
k
For x 1, y 1, Fp,n
x, y Fpk n, we get incomplete Fibonacci p-numbers 7.
k
If x 2, y 1, Fp,n x, y Ppk n, we obtained incomplete Pell p-numbers 8.
k
k
x, y Jn,p1
, we have incomplete generalized
On choosing x 1, y 2, Fp,n
Jacobsthal numbers 4.
k
x, y Fn k, we get incomplete Fibonacci numbers 2.
If x 1, y 1, p 1, Fp,n
k
x, y Fn , we obtained Fibonacci
For x 1, y 1, p 1, k n − 1/p 1, Fp,n
numbers 9.
Definition 2.2. For p ≥ 1, n ≥ 1, incomplete bivariate Lucas p-polynomials are defined as
Lkp,n
k
n
x, y n − jp
j0
n − jp
j
xn−jp1 yj ,
0≤k≤
n
.
p1
2.2
If x 1, y 1, Lkp,n x, y Lkp n, we obtained incomplete Lucas p-numbers 7.
For x 2, y 1, Lkp,n x, y Qpk n, we have incomplete Pell-Lucas p-numbers 8.
k
, we get incomplete generalized
On choosing x 1, y 2, p 1, Lkp,n x, y jn,p1
Jacobsthal-Lucas numbers 4.
If x 1, y 1, p 1, Lkp,n x, y Ln k, we obtained incomplete Lucas numbers 2.
For x 1, y 1, p 1, k n/p 1, Lkp,n x, y Ln , we have Lucas numbers 9.
Proposition 2.3. The incomplete bivariate Fibonacci p-polynomials satisfy the following recurrence
relation:
k1
k1
k
Fp,n
x, y xFp,n−1
x, y yFp,n−p−1
x, y ,
0≤k≤
n−p−3
.
p1
2.3
4
Discrete Dynamics in Nature and Society
Proof. Using 2.1, we obtain
k1
k
x, y yFp,n−p−1
x, y
xFp,n−1
x
k1 n − jp − 2
k1 n − jp − 2
k1 n − jp − 2
k1
k
n − p − pj − 2
xn−p−jp1−2 yj1
y j
j0
n−jp1−1 j
n−jp1−1 j
y k1 n − pj − 2
n − jp − 2
j
n − pj − 2
xn−jp1−1 yj
j −1
j1
j0
x
j
j0
x
j
j0
x
j
j0
k
n − p − pj − 2
xn−p−jp1−2 yj
y y
j
j0
n−jp1−2 j
j −1
x
n−jp1−1 j
y −
2.4
n−2
−1
xn−1
k1 n − jp − 1
xn−jp1−1 yj − 0
j
j0
k1
Fp,n
x, y .
Taking x y 1 in 2.3, we could obtain a formula for incomplete Fibonacci pnumbers 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 Fibonacci ppolynomials is
k
Fp,n
x, y k
xFp,n−1
x, y k
yFp,n−p−1
x, y −
n − pk 1 − 2
xn−pk1−k−2 yk1 .
k
2.5
Proof. It is easy to obtain from 2.1 and 2.3.
Proposition 2.5. For 0 ≤ k ≤ n − h − p − 1/p 1, one has
h
h
j0
j
kj
yh−j xj Fp,np
kh
x, y Fp,n
j−1
p1h−p
x, y .
2.6
Discrete Dynamics in Nature and Society
5
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
h1 h 1
j
j0
h1
h
h1 h
j0
y
j
h
j
h
y
j−1
x, y x
h
x, y x, y
j−1
h1
h
j0
j −1
kj
yh1−j xj Fp,np
j−1
x, y
h1
kh1
xh1 Fp,nph
x, y
2.7
kj1 Fp,npj x, y
h
h
kj1 yh−j xj Fp,npj x, y
x, y x
j
j0
−1
kh
yFp,n
kj
h−j j1
kh
yFp,n
p1h−p
kj
yh1−j xj Fp,np
x Fp,np
x, y
h
h1−j j
h
j−1
j−1
j −1
kh
yFp,n
p1h−p
j
j0
kj
yh1−j xj Fp,np
k
yh1 Fp,n−p
x, y
kh1
x, y xFp,n
x, y
p1
h
x, y .
h1
p1h−p
kh1
Fp,n
p1
Proposition 2.6. For n ≥ kp 1 p 2,
h−1
y
j0
xj
k
Fp,n−pj
x, y 1
xh−1
k1
k1
Fp,nh
x, y − xFp,n
x, y .
2.8
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 Lucas p-polynomials is preserved as found in 5 before.
Proposition 2.7. One has
Lkp,n
x, y k
Fp,n1
x, y k−1
pyFp,n−p
x, y ,
n
0≤k≤
.
p1
2.9
6
Discrete Dynamics in Nature and Society
Proof. By 2.1, rewrite the right-hand side of 2.9 as
k
Fp,n1
x, y k−1
pyFp,n−p
x, y k
n − jp
k
n − jp
k
j0
x
j
j0
x
j
j0
k−1 n − p − jp − 1
y py
xn−p−jp1−1 yj
j
j0
n−jp1 j
n−jp1 j
y py
k
n − jp − 1
j −1
j1
n − jp
j
k
n
n
−
jp
j0
n − jp − 1
n − jp
j
j −1
xn−jp1 yj−1
x
n−jp1 j
y −
n−1
xn
−1
xn−jp1 yj
Lkp,n x, y .
2.10
Proposition 2.8. The incomplete bivariate Lucas p-polynomials satisfy the following recurrence relation:
k1
k
Lk1
p,n x, y xLp,n−1 x, y yLp,n−p−1 x, y ,
0≤k≤
n−p−2
.
p1
2.11
Proof. We write by using 2.3 and 2.9
k1
k
Lk1
p,n x, y Fp,n1 x, y pyFp,n−p x, y
k1
k
k
k−1
xFp,n
x, y yFp,n−p
x, y py xFp,n−p−1
x, y yFp,n−2p−1
x, y
x
k1
Fp,n
x, y k
pyFp,n−p−1
x, y
y
k
Fp,n−p
x, y k−1
pyFp,n−2p−1
x, y
2.12
k
xLk1
p,n−1 x, y yLp,n−p−1 x, y .
Proposition 2.9. The nonhomogeneous recurrence relation of incomplete bivariate Lucas ppolynomials is
Lkp,n x, y xLkp,n−1 x, y yLkp,n−p−1 x, y
n−p−1
−
n − pk 1 − 1
n − pk 1 − 1
k
Proof. The proof can be done by using 2.2 and 2.11.
xn−p1k1 yk1 .
2.13
Discrete Dynamics in Nature and Society
7
Proposition 2.10. For 0 ≤ k ≤ n − p − h/p 1, one has
h
h
j
j0
kj
xj yh−j Lp,np
x, y .
2.14
k1
Lk1
p,nh x, y − xLp,n x, y .
2.15
x, y Lkh
p,n
j−1
p1h−p
Proof. Proof is similar to the proof of Proposition 2.5.
Proposition 2.11. For n ≥ k 1p 1, one has
h−1
y
j0
Lkp,n−pj x, y xj
1
xh−1
Proof. Proof is obtained immediately by using 2.11 and induction h.
Proposition 2.12. One has
n/p1
k0
Lkp,n x, y n
n 1 Lp,n x, y xFp,n x, y − Lp,n x, y .
p1
p1
2.16
Proof. We can write from 2.2
n/p1
k0
n/p1 Lkp,n x, y L0p,n x, y L1p,n x, y L2p,n x, y · · · Lp,n
x, y
n−p
n n
n
n
n−p1
x x x
y
n 0
n−p
0
1
n−p
n − 2p
n
n
n n
n
n−p1
n−2p1 2
x x
x
y
y ···
n 0
n−p
n − 2p
1
2
n
n
n
⎡
⎢n
⎢
⎣n
n
n
0
xn · · · n
n − n/ p 1 p
⎞
⎤
n
⎜n − p 1 p⎟ n−p1n/p1 n/p1 ⎥
⎥
⎟x
×⎜
y
⎝
⎠
⎦
n
p1
⎛
8
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n
n−p
n
n
n
n
n
1
x 1 − 1
xn−p1 y
p1
n 0
p1
n−p
1
n − 2p
n
n
xn−2p1 y2 · · ·
1−2
p1
n − 2p
1
⎞
n
p⎟
n
−
⎜
n
p 1 ⎟xn−p1n/p1 yn/p1
⎜
⎠
n
n − n/ p 1 p ⎝
p1
n/p1
n n − jp
n
1−j
xn−jp1 yj
p
1
n
−
jp
j
j0
⎛
−
n/p1
n − jp
n
n
1
xn−jp1 yj
p1
n − jp
j
j0
n − jp
n
xn−jp1 yj .
j
n − jp
j
n/p1
j0
2.17
Equation 2.17 is calculated using the formula Lp,n x, y and ∂Lp,n x, y/∂x nFp,n x, y 5
n/p1
k0
Lkp,n
x, y n/p1
n − jp
n
n
xn−jp1 yj
1
p1
n
−
jp
j
j0
nFp,n x, y − nx−1 Lp,n x, y
p 1 x−1
n
n 1 Lp,n x, y xFp,n x, y − Lp,n x, y .
p1
p1
2.18
Then we have the following conclusion.
Conclusion 1. When x y p 1 in 2.16, we obtain
n/2
Ln k k0
which is Proposition 11 in 2.
! n "
n
1 Ln Fn − Ln 2
2
2.19
Discrete Dynamics in Nature and Society
9
3. Generating Functions of the Incomplete Bivariate Fibonacci and
Lucas p-Polynomials
Lemma 3.1 see 3. Let {sn }∞
n0 be a complex sequence satisfying the following nonhomogeneous
recurrence relation:
sn xsn−1 ysn−p−1 rn ,
n > p,
3.1
where {rn } is a given complex sequence. Then the generating function Skp x, y; t of the sequence {sn }
is
Skp
p
−1
i
x, y; t s0 − r0 si − xsi−1 − ri t Gt 1 − xt − ytp1 ,
3.2
i1
where Gt denotes the generating function of {rn }.
Theorem 3.2. The generating function of the incomplete bivariate Fibonacci p-polynomials is
Rkp
x, y; t t
kp11
p
Fp,kp11 x, y ti Fp,kp11i x, y − xFp,kp1i x, y
yk1 tp1
1 − xtk1
i1
3.3
−1
1 − xt − ytp1 .
k
Proof. From 2.1 and 2.5, Fp,n
x, y 0 for 0 ≤ n < kp 1 1,
k
Fp,k
p11
k
Fp,k
p12
x, y Fp,kp11 x, y ,
x, y Fp,kp12 x, y ,
3.4
..
.
k
Fp,k
p1p1
x, y Fp,kp1p1 x, y ,
and for n ≥ kp 1 p 2
k
Fp,n
x, y k
xFp,n−1
x, y k
yFp,n−p−1
x, y −
n − pk 1 − 2
xn−pk1−k−2 yk1 . 3.5
n−k p1 −p−2
Now let
k
s0 Fp,k
p11
x, y ,
k
s1 Fp,k
p12
x, y , . . . ,
k
sp Fp,k
p1p1
x, y ,
3.6
10
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and
k
sn Fp,nk
x, y .
p11
3.7
Also
r0 r1 · · · rp 0,
nk−p−1
rn n−p−1
xn−p−1 yk1 .
3.8
We obtained that Gt yk1 tp1 /1 − xtk1 is the generating function of the sequence {rn }.
From Lemma 3.1, we get that the generating function Skp x, y; t of sequence {sn } is
Skp
k
x, y; t Fp,k
p k
x, y ti Fp,k
p11
yk1 tp1
1 − xtk1
i1
1 − xt − yt
p11i
p1
−1
k
x, y − xFp,k
p1i
x, y
3.9
.
Therefore,
Rkp x, y; t tkp11 Skp x, y; t .
3.10
Theorem 3.3. The generating function of the incomplete bivariate Lucas p-polynomials is
Wpk
x, y; t t
kp1
Lp,kp1
p
x, y ti Lp,kp1i x, y − xLp,kp1i−1 x, y
i1
tp1 yk1 p1 − xt 1 1 − xtk1
1 − xt − ytp1
−1
3.11
.
Proof. From 2.9 and 3.3,
∞
Lkp,n x, y tn
Wpk x, y; t n0
∞ k
k−1
Fp,n1
x, y pyFp,n−p
x, y tn
n0
∞
∞
k
k−1
Fp,n1
x, y tn py Fp,n−p
x, y tn
n0
n0
t−1 Rkp x, y; t pytp Rk−1
x, y; t .
p
3.12
Discrete Dynamics in Nature and Society
11
For the general case in Theorems 3.2 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.
References
1 G. B. Djordjević, “Generating functions of the incomplete generalized Fibonacci and generalized Lucas
numbers,” The Fibonacci Quarterly, vol. 42, no. 2, pp. 106–113, 2004.
2 P. Filipponi, “Incomplete Fibonacci and Lucas numbers,” Rendiconti del Circolo Matematico di Palermo.
Serie II, vol. 45, no. 1, pp. 37–56, 1996.
3 Á. Pintér and H. M. Srivastava, “Generating functions of the incomplete Fibonacci and Lucas numbers,” Rendiconti del Circolo Matematico di Palermo. Serie II, vol. 48, no. 3, pp. 591–596, 1999.
4 G. B. Djordjević and H. M. Srivastava, “Incomplete generalized Jacobsthal and Jacobsthal-Lucas numbers,” Mathematical and Computer Modelling, vol. 42, no. 9-10, pp. 1049–1056, 2005.
5 N. Tuglu, E. G. Kocer, and A. Stakhov, “Bivariate Fibonacci like p-polynomials,” Applied Mathematics
and Computation, vol. 217, no. 24, pp. 10239–10246, 2011.
6 E. G. Kocer and N. Tuglu, “The Binet formulas for the Pell and Pell-Lucas p-numbers,” Ars Combinatoria, vol. 85, pp. 3–17, 2007.
7 D. Tasci and M. C. Firengiz, “Incomplete Fibonacci and Lucas p-numbers,” Mathematical and Computer
Modelling, vol. 52, no. 9-10, pp. 1763–1770, 2010.
8 D. Tasci, M. Cetin Firengiz, and G. B. Djordjević, “Incomplete Pell and Pell-Lucas p-numbers,” Submit.
9 T. Koshy, Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics, Wiley, New
York, NY, USA, 2001.
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