On Some Properties of Bivariate Fibonacci and Lucas Polynomials

Hacène Belbachir^{1} and Farid Bencherif^{2} USTHB/Faculty of Mathematics P. O. Box 32 El Alia, 16111 Bab Ezzouar Algeria

                      hbelbachir@usthb.dz
                   hacenebelbachir@gmail.com
                      fbencherif@usthb.dz
                     fbencherif@gmail.com

Abstract In this paper we generalize to bivariate Fibonacci and Lucas polynomials, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate bases are families of integers satisfying remarkable recurrence relations.

1 Introduction

In [4], the authors established that Chebyshev polynomials of the first and second kind admit remarkable integer coordinates in a specific basis. It turns out that this property can be extended to Jacobsthal polynomials [6, 7], Vieta polynomials [18, 10, 14, 15], Morgan-Voyce polynomials [13, 2, 9, 11, 1, 17, 5] and quasi-Morgan-Voyce polynomials [8], and more generally to bivariate polynomials associated with recurrence sequences of order two.

1This research is partially supported by the LAID3 Laboratory.

2This research is partially supported by the LATN Laboratory.

The bivariate polynomials of Fibonacci and Lucas, denoted respectively by (U_{n}) = (U_{n}(x, y)) and (V_{n}) = (V_{n}(x, y)), are polynomials belonging to ℤ[x, y] and defined by

\{ \begin{matrix}U_{0} = 0, U_{1} = 1, \\ U_{n} = xU_{n−1} + yU_{n−2} (n ≥ 2) , \end{matrix} and \{ \begin{matrix}V_{0} = 2, V_{1} = x, \\ V_{n} = xV_{n−1} + yV_{n−2} (n ≥ 2) . \end{matrix}

It is established, see for example [12, 16, 3], that

U_{n+1} = ∑_{k=0}^{[n/2]} \binom{n − k}{k}x^{n−2k}y^{k},  (1)

V_{n} = ∑_{k=0}^{[n/2]} \frac{n}{n − k} \binom{n − k}{k}x^{n−2k}y^{k} (n ≥ 1) .  (2)

Let E_{n} be the ℚ-vector space spanned by the free family C_{n} = (x^{n−2k}y^{k})_{k}, (0 ≤ k ≤ ⌊n/2⌋). Thus the relations (1) and (2) appear as the decompositions of U_{n+1} and V_{n} over the canonical basis C_{n} of E_{n}.

The goal of this paper is to prove that the families 𝔘_{n} := (x^{k}U_{n+1−k})_{k} and 𝔙_{n} := (x^{k}V_{n−k})_{k} for n − 2 ⌊n/2⌋ ≤ k ≤ n − ⌊n/2⌋ constitute two other bases of E_{n} (Theorem 2.1) with respect to which, the polynomials 2U_{n+1} and 2V_{n} admit remarkable integer coordinates.

2 Main results

Theorem 2.1. For any n ≥ 1, 𝔘_{n} and 𝔙_{n} are bases of E_{n}

As U_{n+1} and V_{n} belong to E_{n}, the polynomials U_{2n+1} and V_{2n} are elements of E_{2n} with basis 𝔘_{2n} or 𝔙_{2n}. Similarly, U_{2n} and V_{2n−1} belong to E_{2n−1} with basis 𝔘_{2n−1} or 𝔙_{2n−1}.

Therefore, there are a priori 8 possible decompositions:

over 𝔘_{2n}  1 → trivial,  over 𝔘_{2n}  3 → simple,
U_{2n+1} \begin{matrix}↗ \\ ↘ \end{matrix}  V_{2n} \begin{matrix}↗ \\ ↘ \end{matrix}
over 𝔙_{2n}  2 → T h. A,  over 𝔙_{2n}  4 → trivial,
over 𝔘_{2n−1}  5 → T h. B,  over 𝔘_{2n−1}  7 → T h. C,
U_{2n} \begin{matrix}↗ \\ ↘ \end{matrix}  V_{2n−1} \begin{matrix}↗ \\ ↘ \end{matrix}
over 𝔙_{2n−1}  6 → T h. E,  over 𝔙_{2n−1}  8 → T h. D,

where the cases 1 and 4 are obvious since U_{2n+1} ∈ 𝔘_{2n} and V_{2n} ∈ 𝔙_{2n}.

The decomposition of V_{2n} in 𝔘_{2n} is simple: we have V_{2n} = 2U_{2n+1} − xU_{2n}.

The remaining cases are established by the five following results.

Theorem 2.2. (A). Decomposition of 2U_{2n+1} over the basis 𝔙_{2n}. For every integer n ≥ 0, one has

2U_{2n+1} = ∑_{k=0}^{n} a_{n,k}x^{k}V_{2n−k},

where

a_{n,k} = ∑_{j=0}^{n}(−1)^{j+k}(2 − δ_{n,j})\binom{j}{k}.

Moreover, (a_{n,k})_{n,k≥0} is a family of integers satisfying the following recurrence relation:

\{ \begin{matrix}a_{n,k} = −a_{n−1,k} + a_{n−1,k−1} (n ≥ 1, k ≥ 1); \\ a_{n,0} = 1 (n ≥ 0) ; \\ a_{0,k} = δ_{k,0} (k ≥ 0) . \end{matrix}

(δ_{i,j} being the Kronecker symbol).

The recurrence relation permits us to obtain the following table:

n \ k 0 1 2 3  4  5 6 7 8
0 1
1 1 1
2 1 0 1
3 1 1 −1 1
4 1 0 2 −2  1
5 1 1 −2 4 −3  1
6 1 0 3 −6  7 −4 1
7 1 1 −3 9 −13 11 −5 1
8 1 0 4 −12 22 −24 16 −6 1

from which it follows that

2U_{1} = V_{0},
2U_{3} = V_{2} + xV_{1},
2U_{5} = V_{4} + 0V_{3} + x^{2}V_{2},
2U_{7} = V_{6} + xV_{5} − x^{2}V_{4} + x^{3}V_{3}.

Theorem 2.3. (B). Decomposition of U_{2n} over the basis 𝔘_{2n−1}. For every integer n ≥ 1, one has

U_{2n} = ∑_{k=1}^{n} b_{n,k}x^{k}U_{2n−k}

where

b_{n,k} = (−1)^{k+1}\binom{n}{k}.

Moreover, (b_{n,k})_{n,k≥0} is a family of integers satisfying the following recurrence relation:

\{ \begin{matrix}b_{n,k} = b_{n−1,k} − b_{n−1,k−1} (n ≥ 1, k ≥ 1); \\ b_{n,0} = −1 (n ≥ 0) ; \\ b_{0,k} = −δ_{k,0} (k ≥ 0). \end{matrix}

The latter recurrence relation permits us to obtain the following table:

n \ k 0 1 2  3  4  5  6 7 8
0 −1
1 −1 1
2 −1 2 −1
3 −1 3 −3 1
4 −1 4 −6 4 −1
5 −1 5 −10 10 −5 1
6 −1 6 −15 20 −15 6 −1
7 −1 7 −21 35 −35 21 −7 1
8 −1 8 −28 56 −70 56 −28 8 −1

from which it follows that

U_{2} = xU_{1}
U_{4} = 2xU_{3} − x^{2}U_{2}
U_{6} = 3xU_{5} − 3x^{2}U_{4} + x^{3}U_{3}
U_{8} = 4xU_{7} − 6x^{2}U_{6} + 4x^{3}U_{5} − x^{4}U_{4}

Theorem 2.4. (C). Decomposition of V_{2n−1} over the basis 𝔘_{2n−1}. For every integer n ≥ 1, one has

V_{2n−1} = ∑_{k=1}^{n} c_{n,k}x^{k}U_{2n−k}

where

c_{n,k} = 2 (−1)^{k+1} \binom{n}{k} − δ_{k,1}.

Moreover, (c_{n,k})_{n≥1,k≥0} is a family of integers satisfying the following recurrence relation:

\{ \begin{matrix}c_{n,k} = c_{n−1,k} − c_{n−1,k−1} − δ_{k,2} (n ≥ 2, k ≥ 1); \\ c_{n,0} = −2 (n ≥ 1) ; \\ c_{1,k} = −2δ_{k,0} + δ_{k,1} (k ≥ 0). \end{matrix}

The latter recurrence relation permits us to obtain the following table:

n \ k 0 1  2  3  4  5  6  7 8
1 −2 1
2 −2 3 −2
3 −2 5 −6 2
4 −2 7 −12 8 −2
5 −2 9 −20 20 −10 2
6 −2 11 −30 40 −30 12 −2
7 −2 13 −42 70 −70 42 −14 2
8 −2 15 −56 112 −140 112 −56 16 −2

from which we get

\{ \begin{matrix}V_{1} = xU_{1} \\ V_{3} = 3xU_{3} − 2x^{2}U_{2} \\ V_{5} = 5xU_{5} − 6x^{2}U_{4} + 2x^{3}U_{3} \\ V_{7} = 7xU_{7} − 12x^{2}U_{6} + 8x^{3}U_{5} − 2x^{4}U_{4} \end{matrix}

Theorem 2.5. (D). Decomposition of 2V_{2n−1} over the basis 𝔙_{2n−1}. For every integer n ≥ 1, one has

2V_{2n−1} = ∑_{k=1}^{n} d_{n,k}x^{k}V_{2n−1−k}

where

d_{n,k} = (−1)^{k+1} \frac{2n − k}{n} \binom{n}{k}.

Moreover, (d_{n,k})_{n≥1, k≥0} is a family of integers satisfying the following recurrence relation:

\{ \begin{matrix}d_{n,k} = d_{n−1,k} − d_{n−1,k−1} (n ≥ 2, k ≥ 1); \\ d_{n,0} = −2 (n ≥ 1) ; \\ d_{1,k} = −2δ_{k,0} + δ_{k,1} (k ≥ 0). \end{matrix}

The latter recurrence relation permits us to obtain the following table:

n \ k 0 1  2  3  4  5  6 7 8
1 −2 1
2 −2 3 −1
3 −2 5 −4 1
4 −2 7 −9 5 −1
5 −2 9 −16 14 −6  1
6 −2 11 −25 30 −20 7 −1
7 −2 13 −36 55 −50 27 −8 1
8 −2 15 −49 91 −105 77 −35 9 −1

from which we obtain

\{ \begin{matrix}2V_{1} = xV_{0} \\ 2V_{3} = 3xV_{2} − x^{2}V_{1} \\ 2V_{5} = 5xV_{4} − 4x^{2}V_{3} + x^{3}V_{2} \\ 2V_{7} = 7xV_{6} − 9x^{2}V_{5} + 5x^{3}V_{4} − x^{4}V_{3} \end{matrix}

Theorem 2.6. (E). Decomposition of 2U_{2n} over the basis 𝔙_{2n−1}. For every integer n ≥ 1, one has

2U_{2n} = ∑_{k=1}^{n} e_{n,k}x^{k}V_{2n−1−k}

where

e_{n,k} = (−1)^{k+1} \frac{2n − k}{2n} \binom{n}{k}+δ_{k,0} + \frac{1}{2} ∑_{j=0}^{n−1}(−1)^{j+k−1}(2 − δ_{n−1,j})\binom{j}{k − 1}.

Moreover, (e_{n,k})_{n,k≥0} is a family of integers satisfying the following recurrence relation:

\{ \begin{matrix}e_{n,k} = e_{n−2,k} − 2e_{n−2,k−1} + e_{n−2,k−2} (n ≥ 3, k ≥ 2); \\ e_{n,0} = 0 and e_{n,1} = n (n ≥ 1) ; \\ e_{1,k} = δ_{k,1} and e_{2,k} = 2δ_{k,1} (k ≥ 0). \end{matrix}

The latter recurrence relation permits us to obtain the following table:

n \ k 0 1 2  3  4  5  6 7 8
1  0 1
2  0 2 0
3  0 3 −2 1
4  0 4 −4 2 0
5  0 5 −8 8 −4 1
6  0 6 −12 14 −8 2 0
7  0 7 −18 29 −28 17 −6 1
8  0 8 −24 44 −48 32 −12 2 0

from which, we have

\{ \begin{matrix}2U_{2} = xV_{0} \\ 2U_{4} = 2xV_{2} + 0x^{2}V_{0} \\ 2U_{6} = 3xV_{4} − 2x^{2}V_{3} + x^{3}V_{2} \\ 2U_{8} = 4xV_{6} − 4x^{2}V_{5} + 2x^{3}V_{4} + 0x^{4}V_{3} \end{matrix}

3 Proof of Theorems

Theorem 1 follows from the following lemma.

Lemma 3.1. For any integer n ≥ 0, by setting m = ⌊n/2⌋ , we have

det_{C_{n}}(𝔘_{n}) = (−1)^{m(m+1)/2} and det_{C_{n}}(𝔙_{n}) = 2(−1)^{m(m+1)/2}.

Proof. Let us prove only the first equality as the proof of the other one is similar.

Let r = n−2m, W_{k}^{(m)} = x^{k}U_{2m+1−k} (0 ≤ k ≤ m) and Δ_{m} = det_{C_{2m}}(W_{0}^{(m)}, W_{1}^{(m)}, …, W_{m}^{(m)}), we have

det_{C_{n}}(𝔘_{n}) = det_{C_{2m+r}} (x^{r+k}U_{2m+1−k})_{0≤k≤m} = Δ_{m}.

The result follows by noticing that Δ_{0} = 1 and Δ_{m} = (−1)^{m}Δ_{m−1} for m ≥ 1. Indeed, for m ≥ 1, we have

W_{k+1}^{(m)} − W_{k}^{(m)} = x^{k}(xU_{2m−k} − U_{2m−k+1}) = −yW_{k}^{(m−1)} (0 ≤ k ≤ m − 1) .

Thus,

Δ_{m} = det_{C_{2m}}(W_{0}^{(m)}, W_{1}^{(m)} − W_{0}^{(m)}, ..., W_{m−1}^{(m)} − W_{m−2}^{(m)}, W_{m}^{(m)} − W_{m−1}^{(m)})
= det_{C_{2m}}(W_{0}^{(m)}, −yW_{0}^{(m−1)}, −yW_{1}^{(m−1)}, …, −yW_{m−1}^{(m−1)}).

The “component” of W_{0}^{(m)} = U_{2m+1} over x^{2m} is equal to 1. The “component” of −yW_{k}^{(m−1)} over x^{2n}, is equal to 0, for 1 ≤ k ≤ m , so we have

Δ_{m} = det_{C_{2m−2}}(−W_{0}^{(m−1)}, −W_{1}^{(m−1)}, …, −W_{m−1}^{(m−1)}) = (−1)^{m}Δ_{m−1}.

Let A_{m}, B_{m}, C_{m}, D_{m} and E_{m} be the operators on (ℚ [x, y])^{ℕ} defined by

A_{m} = − (x − E)^{m} + 2 ∑_{k=0}^{m} E^{k} (x − E)^{m−k} (m ≥ 0) ,
B_{m} = − (E − x)^{m} (m ≥ 0) ,
C_{m} = 2E^{m} + 2B_{m} − xE^{m−1} (m ≥ 1) ,
D_{m} = (E − x)^{m−1} (x − 2E) (m ≥ 1) ,
E_{m} = \frac{1}{2}(xA_{m−1} + D_{m}) + E^{m} (m ≥ 1) ,

where E is the forward shift operator given by

E ((W_{n})_{n}) = (W_{n+1})_{n}

Then, we have

A_{m} = ∑_{k=0}^{m} a_{m,k}x^{k}E^{m−k}  with a_{m,k} = ∑_{j=0}^{m}(−1)^{j+k}(2 − δ_{m,j})\binom{j}{k}

B_{m} = ∑_{k=0}^{m} b_{m,k}x^{k}E^{m−k}  with b_{m,k} = (−1)^{k+1} \binom{m}{k}

C_{m} = ∑_{k=1}^{m} c_{m,k}x^{k}E^{m−k}  with c_{m,k} = 2 (−1)^{k+1} \binom{m}{k} − δ_{k,1}

D_{m} = ∑_{k=0}^{m} d_{m,k}x^{k}E^{m−k}  with d_{m,k} = (−1)^{k+1} \frac{2m − k}{m} \binom{m}{k}

E_{m} = ∑_{k=1}^{m} e_{m,k}x^{k}E^{m−k}  with e_{m,k} = \frac{1}{2} (d_{m,k}+a_{m−1,k−1}) + δ_{k,0}.

With these notations, relations stated by Theorems A, B, C, D and E may be expressed by means of the following relations

a. ∀n ∈ ℕ A_{n}V_{n} = 2U_{2n+1}
b. ∀n ∈ ℕ^{∗} B_{n}U_{n} = 0
c. ∀n ∈ ℕ^{∗} C_{n}U_{n} = V_{2n−1}
d. ∀n ∈ ℕ^{∗} D_{n}V_{n−1} = 0
e. ∀n ∈ ℕ^{∗} E_{n}V_{n−1} = 2U_{n}

which are to be proven. For this, the following lemma will be useful for us. Lemma 3.2. For every integers n and m, we have

1. V_{n} = 2U_{n+1} − xU_{n} (n ≥ 0) and V_{n} = U_{n+1} + yU_{n−1} (n ≥ 1) ,

2. (E − x)^{n} U_{m} = y^{n}U_{m−n} and (E − x)^{n} V_{m} = y^{n}V_{m−n} (m ≥ n ≥ 0) ,

3. ∑_{k=1}^{n} (−y)^{n−k} V_{2k} = U_{2n+1} − (−y)^{n} (n ≥ 0) .

Proof.

1. See relation (2.9) and (2.8) in [16].

2. We proceed by induction on n.

3. For every integer n ∈ ℕ, put T_{n} := U_{2n+1}−∑_{k=1}^{n} (−y)^{n−k} V_{2k}. The relation to be proven is equivalent to T_{n} = (−y)^{n} (n ≥ 0) . Then, we remark that from the first relation of this lemma, we have for every integer n ≥ 1

T_{n} + yT_{n−1} = U_{2n+1} + yU_{2n−1} − V_{2n} = 0,

(T_{n})_{n≥0} is then a geometric sequence with multiplier (−y) and of first term T_{0} = 1. It follows that for every integer n ∈ ℕ, T_{n} = (−y)^{n} .

Proof of relations a., b., c., d. and e. Using the above Lemma, we have a. A_{n}V_{n} = (−y)^{n} V_{0} + 2 ∑_{k=1}^{n} (−y)^{n−k} V_{2k} = 2U_{2n+1}. b. B_{n}U_{n} = − (E − x)^{n} U_{n} = −y^{n}U_{0} = 0. c. C_{n}U_{n} = (2E^{n} + 2B_{n} − xE^{n−1}) U_{n} = 2U_{2n} − xU_{2n−1} = V_{2n−1}. d. D_{n}V_{n−1} = (E − x)^{n−1} (xV_{n−1} − 2V_{n}) = y^{n−1} (xV_{0} − 2V_{1}) = 0. e. E_{n}V_{n−1} = (\frac{1}{2}xA_{n−1} + \frac{1}{2}D_{n} + E^{n}) V_{n−1} = \frac{1}{2}xA_{n−1}V_{n−1} + \frac{1}{2}D_{n}V_{n−1} + V_{2n−1}. Using

A_{n−1}V_{n−1} = 2U_{2n−1} and D_{n}V_{n−1} = 0, it follows that E_{n}V_{n−1} = xU_{2n−1} + V_{2n−1} = 2U_{2n} Remark 3.1. Theorems A, B, C, D and E generalize results obtained for the Chebyshev polynomials [4], Indeed,

\frac{1}{2}V_{n}(2x, −1) = T_{n}(x) is the Chebyshev polynomials of the first kind,
U_{n+1}(2x, −1) = U_{n}(x) is the Chebyshev polynomials of the second kind,

with

\{ \begin{matrix}T_{n}(x) = 2xT_{n−1} − T_{n−2}, \\ T_{0} = 1, T_{1} = x, \end{matrix} and \{ \begin{matrix}U_{n}(x) = 2xU_{n−1} − U_{n−2}, \\ U_{0}(x) = 1, U_{1} = 2x. \end{matrix}

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2000 Mathematics Subject Classification: Primary 11B39; Secondary 11B37 . Keywords: bivariate Fibonacci and Lucas polynomials, linear recurrences.

(Concerned with sequences \underline{A007318}, \underline{A029653}, and \underline{A112468}.)

Received October 8 2007; revised version received June 3 2008. Published in Journal of Integer Sequences, June 27 2008.

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