Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger’s

Question

Hacène Belbachir

University of Sciences and Technology Houari Boumediene Faculty of Mathematics P. O. Box 32, El Alia

Bab-Ezzouar 16111, Algiers

Algeria

hacenebelbachir@gmail.com

Athmane Benmezai

Faculty of Economy and Management Science

University of Dely Brahim Rue Ahmed Ouaked Dely Brahim, Algiers,

Algeria

athmanebenmezai@gmail.com

Abstract

We give an answer to a recent question of Prodinger, which consists of finding q-analogues of identities related to Fibonacci and Lucas polynomials.

1 Introduction

The bivariate polynomials of Fibonacci and Lucas, denoted respectively by (U_{n}) and (V_{n}), are defined by

\{ \begin{matrix}U_{0} = 0, U_{1} = 1, \\ U_{n} = tU_{n−1} + zU_{n−2} (n ≥ 2) , \end{matrix} and \{ \begin{matrix}V_{0} = 2, V_{1} = t, \\ V_{n} = tV_{n−1} + zV_{n−2} (n ≥ 2) . \end{matrix}

It is well-known (see, for example, [3]) that

U_{n+1} = ∑_{k=0}^{⌊n/2⌋} \binom{n − k}{k}t^{n−2k}z^{k}, V_{n} = ∑_{k=0}^{⌊n/2⌋} \frac{n}{n − k} \binom{n − k}{k}t^{n−2k}z^{k} (n ≥ 1) .

In [1], Belbachir and Bencherif proved the following five formulae:

2U_{2n+1} = ∑_{k=0}^{n} a_{n,k}t^{k}V_{2n−k}, with a_{n,k} = 2 ∑_{j=0}^{n} (−1)^{j+k} \binom{j}{k} − (−1)^{n+k} \binom{n}{k},  (1)

V_{2n} = ∑_{k=1}^{n} b_{n,k}t^{k}V_{2n−k}, with b_{n,k} = (−1)^{k+1} \binom{n}{k},  (2)

V_{2n−1} = ∑_{k=1}^{n} c_{n,k}t^{k}U_{2n−k}, with c_{n,k} = 2 (−1)^{k+1} \binom{n}{k} − [k = 1] ,  (3)

2V_{2n−1} = ∑_{k=1}^{n} d_{n,k}t^{k}V_{2n−1−k}, with d_{n,k} = (−1)^{k+1} \frac{2n − k}{n} \binom{n}{k},  (4)

2U_{2n} = ∑_{k=1}^{n} e_{n,k}t^{k}V_{2n−k}, with  (5)

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

and they also gave the following simple sixth one:

V_{2n} = 2U_{2n+1} − xU_{2n}.  (6)

The results presented in [1] are companion generalizations for Chebyshev polynomials [2].

Without loss of generality, we suppose that t = 1. We refer here to the modified polynomials given by Prodinger in the introduction of [5].

In order to give q-analogues of these instances, Prodinger considered Cigler’s [4] suggestion to replace U_{n} and V_{n}, respectively, with

F_{n+1} (z) = ∑_{k=0}^{⌊n/2⌋} q^{\binom{k+1}{2}}\left[{n − k \atop k}\right]_{q}z^{k}, L_{n} (z) = ∑_{k=0}^{⌊n/2⌋} q^{\binom{k}{2}}\left[{n − k \atop k}\right]_{q} \frac{[n]_{q}}{[n − k]_{q}} z^{k},

with the following q-notations

[n]_{q} = 1 + q + ⋯ + q^{n−1}, [n]_{q}! = [1]_{q}[2]_{q} ⋯ [n]_{q}, \left[{n \atop k}\right]_{q} = \frac{[n]_{q}!}{[k]_{q}![n − k]_{q}!}.

Prodinger [5] found the following two q-analogues for identities (4) and (6) respectively

F_{2n} (z) = ∑_{k=1}^{n} β_{n,k}F_{2n−k} (z) , with β_{n,k} = (−1)^{k+1} q^{\binom{k}{2}}\left[{n \atop k}\right]_{q},  (7)

L_{2n−1} = ∑_{k=1}^{n} δ_{n,k}L_{2n−1−k} (z) , with δ_{n,k} = (−1)^{k+1} \frac{q^{\binom{k}{2}}}{1 + q^{n−1}} (\left[{n − 1 \atop k}\right]_{q} + q^{n−1}\left[{n \atop k}\right]_{q}) ,(8)

and left the three others as a challenge.

We give an answer to the remaining three instances and we find a q-analogue for identity (6).

2 An answer to Prodinger’s question

Here, we find a q-analogue for the identities (6), (3), (1) and (5), which can be considered as an answer to Prodinger’s question [5].

Theorem 1. The following q-identities hold

L_{2n} (z) = F_{2n+1} (\frac{z}{q} ) + F_{2n+1} (z) − F_{2n} (z) ,  (9)
L_{2n−1} (z) = ∑_{k=1}^{n} β_{n,k} (F_{2n−k} (z) + F_{2n−k} (\frac{z}{q} )) − F_{2n−1} (z) ,  (10)
F_{2n+1} (z) = \frac{1}{1 − q^{2n}} (L_{2n} (qz) − q^{2n}L_{2n} (z)) ,  (11)
F_{2n} (z) = \frac{1}{1 − q^{2n−1}} (L_{2n−1} (qz) − q^{2n−1}L_{2n−1} (z)) .  (12)

Proof. We have

F_{n+1} (\frac{z}{q} ) + F_{n+1} (z) − F_{n} (z) = ∑_{k=0}^{⌊n/2⌋} q^{\binom{k}{2}}\left[{n − k \atop k}\right]_{q} (1 + q^{k} − q^{k} \begin{matrix}[n − 2k] & q \\ [n − k] & q \end{matrix} ) z^{k}

= ∑_{k=0}^{⌊n/2⌋} q^{\binom{k}{2}}\left[{n − k \atop k}\right]_{q} (1 + q^{k} − q^{k} \frac{1 − q^{n−2k}}{1 − q^{n−k}} ) z^{k}

= ∑_{k=0}^{⌊n/2⌋} q^{\binom{k}{2}}\left[{n − k \atop k}\right]_{q} \frac{1 − q^{n}}{1 − q^{n−k}} z^{k}

= ∑_{k=0}^{⌊n/2⌋} q^{\binom{k}{2}}\left[{n − k \atop k}\right]_{q} \frac{[n]_{q}}{[n − k]_{q}} z^{k} = L_{n} (z) .

For the second identity ∑_{k=1}^{n} β_{n,k} (F_{2n−k} (z) + F_{2n−k} (\frac{z}{q}))−F_{2n−1} (z) = F_{2n} (z)+F_{2n} (\frac{z}{q})− F_{2n−1} (z) = L_{2n−1} (z) .

To establish (11) and (12), we have

L_{n} (qz) − q^{n}L_{n} (z) = ∑_{k=0}^{⌊n/2⌋} q^{\binom{k}{2}}\left[{n − k \atop k}\right]_{q} \begin{matrix}[n] & q \\ [n − k] & q \end{matrix} (q^{k} − q^{n}) z^{k}
= ∑_{k=0}^{⌊n/2⌋} q^{\binom{k+1}{2}}\left[{n − k \atop k}\right]_{q} (1 − q^{n}) z^{k}
= (1 − q^{n}) F_{n+1} (z) ,

and then take n odd and n even respectively.

Remark 2. We have to notice that the expressions of relations (11) and (12) do not give, for q = 1, the initial formulae as given in relations (1) and (5). We obtain, when q → 1, the well known identity U_{n+1} (z) = V_{n} (z) − zU_{n−1} (z) , for n odd and n even respectively. However, they express, as in Belbachir’s and Bencherif’s work [1], an odd Fibonacci polynomial in terms of Lucas polynomials starting with a smallest even index, and an even Fibonacci polynomial in terms of Lucas polynomials starting with a smallest odd index.

3 Acknowledgment

The authors would like to thank the referee for his patience and very useful guidance which improved the quality of the paper.

References

[1] H. Belbachir and F. Bencherif, On some properties of bivariate Fibonacci and Lucas polynomials, J. Integer Sequences 11 (2008), Article 08.2.6.

[2] H. Belbachir and F. Bencherif, On some properties of Chebyshev polynomials, Discuss. Math. Gen. Algebra Appl 28 (2008), 121–133.

[3] H. Belbachir and F. Bencherif, Linear recurrent sequences and powers of a square matrix. Integers 6 (2006), Paper A12.

[4] J. Cigler, A new class of q-Fibonacci polynomials, Electronic J. Combinatorics 10 (2003), Article R19.

[5] H. Prodinger, On the expansion of Fibonacci and Lucas polynomials, J. Integer Sequences 12 (2009), Article 09.1.6.

2000 Mathematics Subject Classification: Primary 11B39; Secondary 05A30, 11B37. Keywords: Fibonacci polynomials; Lucas polynomials; q-analogues.

(Concerned with sequences \underline{A000032}, \underline{A000045}, \underline{A007318}, \underline{A029653}, and \underline{A112468}.)

Received July 15 2012; revised versions received August 14 2012; September 3 2012. Published in Journal of Integer Sequences, September 8 2012.

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