A note on h(x) − Fibonacci quaternion polynomials

Chaos, Solitons and Fractals 77 (2015) 1–5

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

A note on h(x) − Fibonacci quaternion polynomialsR Paula Catarino∗ Department of Mathematics University of Trás-os-Montes e Alto Douro, UTAD, 5001 - 801 Vila Real, Portugal

a r t i c l e

i n f o

Article history: Received 25 February 2015 Accepted 23 April 2015

a b s t r a c t In this paper, we introduce h(x) − Fibonacci quaternion polynomials that generalize the k − Fibonacci quaternion numbers, which in their turn are a generalization of the Fibonacci quaternion numbers. We also present a Binet-style formula, ordinary generating function and some basic identities for the h(x) − Fibonacci quaternion polynomial sequences.

MSC: 11B39 11R52 05A15

1. Introduction The investigation of normed division algebras is a research topic of great interest. There are exactly four normed division algebras: the real numbers R, complex numbers C, quaternions H, and octonions O. The quaternions H are the noncommutative normed division algebra over the real numbers R and due the commutativity, one cannot directly extend various results on real and complex numbers to quaternions. For general material on quaternions we refer to the book [9]. Nowadays, quaternions play an important role in computer science, quantum physics, signal, and color image processing (see, for example [1]). In addition to this, studies on different types of sequences of quaternions (Fibonacci Quaternions, Split Fibonacci Quaternions and Complex Fibonacci Quaternions) have been done by several researchers (see, [2,3,14,15,17,20]). A great investigation is dedicated to Fibonacci numbers and their generalizations (see, for example, [4,7,8,12,18], among other works) and there is a huge interest in the theory and applications of the golden section and Fibonacci numbers

R Member of CM-UTAD, Collaborator of CIDTFF, Portuguese Research Centers. ∗ Corresponding author. Tel.: +351259350326; fax.: +351259350817. E-mail address: [email protected], [email protected] URL: http://www.utad.pt

http://dx.doi.org/10.1016/j.chaos.2015.04.017 0960-0779/© 2015 Elsevier Ltd. All rights reserved.

© 2015 Elsevier Ltd. All rights reserved.

(see, for example, [5,6,21,23,26,27], among others). For background about the Fibonacci numbers, the reader can find their properties in the books [16] and [28]. An example of generalization of the classical Fibonacci sequence is the sequence of k − Fibonacci numbers Fk, n defined by

Fk,n = kFk,n−1 + Fk,n−2 ,

n ≥ 2,

(1)

with the initial values Fk, 0 = 0 and Fk, 1 = 1. Fibonacci-like recursion relations are a special case of difference equations that can be solved by the combinatorics function technique method. Polynomials that can be defined by Fibonacci-like recursion relations are called Fibonacci polynomials and they were studied in 1883 by E. C. Catalan and E. Jacobsthal. More mathematicians were involved in the study of Fibonacci polynomials, such as P. F. Byrd, M. BicknellJohnson, among others. Catalan studied the polynomials Fn (x) defined by the recurrence relation

Fn (x) = xFn−1 (x) + Fn−2 (x),

n ≥ 3,

where F1 (x) = 1 and F2 (x) = x. Let h(x) be a polynomial with real coefficients. Nalli and Haukkanen, in [24], introduced h(x) − Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials, and also, the k − Fibonacci numbers. In this note, we introduce h(x) − Fibonacci quaternion polynomials that generalize the k − Fibonacci quaternion

2

P. Catarino / Chaos, Solitons and Fractals 77 (2015) 1–5

numbers, which in turn are a generalization of the Fibonacci quaternion numbers, whose definitions are included in the next section. The rest of this note is structured as follows: some background, which is required, is given in Section 2; in Section 3, we introduce the h(x) − Fibonacci quaternion polynomials and derive the Binet-style formula, the generating function and some identities of the h(x) − Fibonacci quaternion polynomial sequence. 2. Auxiliary technical concepts and results In this section, some concepts and results are collected in order to facilitate the presentation and proof of results in the rest of the paper. Definition 2.1. [21]. The classic Fibonacci {Fn }n∈N sequence is defined by

Fn = Fn−1 + Fn−2 ,

n ≥ 2,

with the initial conditions F0 = 0 and F1 = 1. We can extend the classic Fibonacci sequence {Fn }n∈N to a negative n. In fact, still keeping to the rule that a Fibonacci number is the sum of the two numbers on its left, we get

. . ., F−5 = 5,

F−4 = −3,

F−3 = 2,

F−2 = −1,

F−1 = 1, . . . and, in this case, the following recurrence relation is satisfied

Fn = Fn+2 − Fn+1 ,

n < 0.

It can be shown (see, for example, [29]), if we allow negative values of the subscript n, that

F−n = (−1)n−1 Fn . The books [16,21,28] give many results dealing with this sequence. Definition 2.2. [24] Let h(x) be a polynomial with real coefficients. The h(x) − Fibonacci polynomials {Fh,n (x)}∞ n=0 are defined by the recurrence relation

Fh,n+1 (x) = h(x)Fh,n (x) + Fh,n−1 (x),

n ≥ 1,

with the initial conditions Fh, 0 (x) = 0 and Fh, 1 (x) = 1. For h(x) ࣕ k, k any real number, we obtain the k − Fibonacci numbers studied by several researchers (see, for example, [4,7,8,11,12,18]). In particular, for k = 1 and k = 2, we obtain the classic Fibonacci numbers and the Pell numbers, respectively. The Pell sequence has also been studied by several authors (see, for example, [10,13,19,25]). The extension of Fh, n (x) to negative values of the subscript n can be obtained by replacing n by − n in the Binet-style formula of Fh, n (x) given by Theorem 2.3 in [24], and, then, we have

Fh,−n (x) = (−1)n+1 Fh,n (x). The quaternion was formally introduced by W. R. Hamilton in 1843 and some background about these types of hypercomplex numbers can be found, for example, in [30] and [9]. Definition 2.3. A quaternion q is defined by

q = q0 + q1 i1 + q2 i2 + q3 i3 ,

where q0 , q1 , q2 , q3 ∈ R and i1 , i2 and i3 are complex operators such that i21 = i22 = i23 = −1, i1 i2 = −i2 i1 = i3 , i2 i3 = −i3 i2 = i1 , i3 i1 = −i1 i3 = i2 and i1 i2 i3 = −1. The field H ∼ = C2 of quaternions is a four-dimensional noncommutative R−field generated by four base elements e0 ࣅ1, e1 ࣅi1 , e2 ࣅi2 and e3 ࣅi3 . The multiplication satisfies the following rules:

e2l = −1,

l ∈ {1, 2, 3}; em en = −en em = βmn et ,

m = n, m, n ∈ {1, 2, 3},

(2)

where β mn and et are uniquely determined by em and en . Some results on Fibonacci quaternions can be found in [17] and following the terminology used, we define the nth Fibonacci quaternion number as follows: Definition 2.4. The Fibonacci quaternion number of order n is defined by

Qn = Fn + Fn+1 i1 + Fn+2 i2 + Fn+3 i3 , where Fn is the nth Fibonacci number and i1 , i2 , i3 , satisfy the identities stated in the previous Definition and n = 0, ±1, ±2, . . .. The k − Fibonacci quaternion numbers are a generalization of the Fibonacci quaternion numbers. We define the nth k − Fibonacci quaternion number as follows: Definition 2.5. The k − Fibonacci quaternion number of order n is defined by

Qk,n = Fk,n + Fk,n+1 i1 + Fk,n+2 i2 + Fk,n+3 i3 , where Fk, n is the nth k − Fibonacci number and i1 , i2 , i3 satisfy the identities stated in the Definition 2.3 and n = 0, ±1, ±2, . . .. Note that, using (1), we easily obtain

Qk,n = (kFk,n−1 + Fk,n−2 ) + Fk,n+1 i1 + Fk,n+2 i2 + Fk,n+3 i3 , another expression for the k − Fibonacci quaternion number of order n. 3. The h(x) − Fibonacci quaternion polynomials Let ei , i = 0, 1, 2, 3, be a basis of H, which satisfy the multiplication rules (2). Let h(x) be a polynomial with real coefficients. Definition 3.1. The h(x) − Fibonacci quaternion polynomials {Qh,n (x)}∞ n=0 are defined by the recurrence relation

Qh,n (x) =

3 

Fh,n+l (x)el ,

l=0

where Fh, n (x) is the nth h(x) − Fibonacci polynomial. For h(x) ࣕ k, k any real number, we get the k − Fibonacci numbers Fk, n from the h(x) − Fibonacci polynomials Fh, n (x) and, thus, we obtain k − Fibonacci quaternion numbers Qk, n from the h(x) − Fibonacci quaternion polynomials Qh, n (x). Next, we shall give the generating function for the h(x) − Fibonacci quaternion polynomial sequences. We shall write the h(x) − Fibonacci quaternion polynomial sequence as a power series, where each term of the sequence correspond to

P. Catarino / Chaos, Solitons and Fractals 77 (2015) 1–5

coefficients of the series in which we do not have to consider the convergence. For background about generating functions see, for example [31]. Let us consider the h(x) − Fibonacci quaternion polynomial sequences {Qh,n (x)}∞ n=0 . By definition of ordinary generating function of some sequence, considering this sequence, the ordinary generating function associated gQ (t) is defined by

gQ (t) =

∞ 

Qh,n (x)tn .

(3)

n=0

Now, with tools given before, we present the fundamental results of this paper. Theorem 3.2. The ordinary generating function for the h(x) − Fibonacci quaternion polynomials Qh, n (x) is

Qh,0 (x) + (Qh,1 (x) − h(x)Qh,0 (x))t gQ (t) = . 1 − h(x)t − t2

(4)

The Binet formula is also well known for several of sequences defined by recurrence relations. Claude Levesque, in [22], finds the general Binet formula for a general mth order linear recurrence. Sometimes, some basic properties derive from this formula. In [15], the author studied the Fibonacci quaternions and gives their generating function and the respective Binet-style formula. This type of formula can also be carried out for the h(x) − Fibonacci quaternion polynomials. Let r1 (x) and r2 (x) denote the roots of the characteristic equation

r2 − h(x)r − 1 = 0 on the recurrence relation of Definition 2.2. Then,

r1 (x) =

gQ (t) = Qh,0 (x) + Qh,1 (x)t + Qh,2 (x)t2 + · · · + Qh,n (x)tn + · · · .

(5)

Multiplying both sides of (5) by h(x)t, we obtain

h(x)gQ (t)t = h(x)Qh,0 (x)t + h(x)Qh,1 (x)t2 + · · · + h(x)Qh,n (x)tn+1 + · · · .

(6)

Now, multiplying both sides of (5) by t2 , we get

(7)

Note that

r2 (x) r1 (x) = −r1 (x)2 ; = −r2 (x)2 . r2 (x) r1 (x)

(1 − h(x)t − t )gQ (t) = Qh,0 (x) + (Qh,1 (x) − h(x)Qh,0 (x))t 

Note that, using once more the Definition 3.1 and also the Definition 2.2, including the initials conditions, we get

Fh,l+1 (x)el − h(x)

=

The following result, with easy proof, uses the Binet-style formula of Fh, n (x) given by Theorem 2.3 in [24] and it will be useful in the statement of the Binet-style formula of Qh, n (x).

Fh,l+1 (x) − r2 (x)Fh,l (x) = r1l (x); Fh,l+1 (x)

3 

Also, the next result will be useful for stating the Binetstyle formula of the h(x) − Fibonacci quaternion polynomials Qh, n (x). Using the properties (9) and the result of Theorem 3.2, we get Lemma 3.5. For the generating function given in Theorem 3.2, we have

Fh,l (x)el

gQ (t) =

l=0

3 

(Fh,l+1 (x) − h(x)Fh,l (x))el

l=0

= e0 + e2 + h(x)e3 Qh,0 (x) + (e0 + e2 + h(x)e3 )t , 1 − h(x)t − t2

a different way to present the ordinary generating function of Qh, n (x). Now, we present the generating function of the h(x) − Fibonacci quaternion polynomials Qh, n (x) when h(x) is an odd polynomial. Theorem 3.3. If h(x) is an odd polynomial then gQ (t) = ∞ n n=0 Qh,n (−x)(−t) and we have

gQ (t) =

Qh,0 (−x) − (Qh,1 (−x) + h(x)Qh,0 (−x))t . 1 + h(x)t − t2



Qh,1 (x) − r2 (x)Qh,0 (x) 1 − r1 (x)t  Qh,1 (x) − r1 (x)Qh,0 (x) − . 1 − r2 (x)t

1 r1 (x) − r2 (x)

Proof. From the expression of gQ (t) in Theorem 3.2 and the use of the properties (9), we have:

and, then,

gQ (t) =

(10)

− r1 (x)Fh,l (x) = r2l (x).

2

l=0

(9)

and

From Definition 3.1, (3), (6) and (7), we have

Qh,1 (x) − h(x)Qh,0 (x) =

  h(x) − h2 (x) + 4 h2 (x) + 4 ; r2 (x) = . 2 2

Lemma 3.4. For the h(x) − Fibonacci polynomials Fh, l (x) we have

gQ (t)t2 = Qh,0 (x)t2 + Qh,1 (x)t3 + · · · + Qh,n (x)tn+2 + · · · .

3 

h(x) +

r1 (x) + r2 (x) = h(x); r1 (x)r2 (x) = −1; r1 (x) − r2 (x)  = h2 (x) + 4

Proof. Using (3), we have

and the result follows.

3

(8)

gQ (t) =

Qh,0 (x) + (Qh,1 (x) − h(x)Qh,0 (x))t 1 − h(x)t − t2

=

Qh,0 (x) + (Qh,1 (x) − h(x)Qh,0 (x))t 1 − (r1 (x) + r2 (x))t + r1 (x)r2 (x)t2

=

Qh,0 (x) + (Qh,1 (x) − h(x)Qh,0 (x))t (1 − r1 (x)t)(1 − r2 (x)t)

and multiplying and dividing the right side of gQ (t) by r1 (x) − r2 (x) the result follows.  The following result gives us the Binet-style formula for Qh, n (x).

4

P. Catarino / Chaos, Solitons and Fractals 77 (2015) 1–5

Theorem 3.6. For n ࣙ 0,

r∗ (x)r1n (x) − r2∗ (x)r2n (x) Qh,n (x) = 1 , r1 (x) − r2 (x) where r1∗ (x) =

3

l l=0 r1 (x)el

(11) 3

and r2∗ (x) =

Theorem 3.8 (Catalan’s identity). For n and r, nonnegative integer numbers, such that r ࣘ n, we have

l l=0 r2 (x)el .

Proof. Using the Lemma 3.5, we have

gQ (t) =

1 r1 (x) − r2 (x)



(Qh,1 (x) − r2 (x)Qh,0 (x))

− (Qh,1 (x) − r1 (x)Qh,0 (x))



∞ 

r2n

(x)t

n

∞ 

2 Qh,n+r (x)Qh,n−r (x) − Qh,n (x)

r1n (x)tn

n=0

Using the Definition 3.1, we obtain:

−

(−r1∗ (x)r2∗ (x))(r1 (x)r2 (x))n ( rr21r ((xx)) + = (r1 (x) − r2 (x))2 r

n=0 ∞ 

l=0

n=0

(Fh,l+1 (x) − r1 (x)Fh,l (x))el

=

 r2n

(x)t

n

.

Using the Lemma 3.4, we get

 3 ∞   1 gQ (t) = r1l (x)el r1n (x)tn r1 (x) − r2 (x) n=0 l=0  3 ∞   − r2l (x)el r2n (x)tn . l=0

For r1∗ (x) =

gQ (t) =

3

∞   n=0

and r2∗ (x) =

(r1∗ (x)r2∗ (x))(−1)n (r1r (x) − r2r (x))2 . (r1 (x) − r2 (x))2

 Note that, for r = 1 in Catalan’s identity obtained, we get the Cassini identity for the h(x) − Fibonacci quaternion polynomials. In fact, the equation of Theorem 3.8, for r = 1, yields,

l l=0 r2 (x)el ,

we obtain



r1∗ (x)r1n (x) − r2∗ (x)r2n (x) n t r1 (x) − r2 (x) 

The d’Ocagne identity can also obtained using the Binet-style formula. The next result for the h(x) − Fibonacci quaternion polynomials is easily proved using the Binet-style formula and, again, some identities of (9) and (10) involving the roots r1 (x) and r2 (x). Theorem 3.10 (d’Ocagne’s identity). Suppose that n is a nonnegative integer number and m any natural number. If m > n, then

For m any integer number and n any natural number, the next result is related with generating function of h(x) − Fibonacci quaternion polynomials, stating the following result:

Qh,m (x)Qh,n+1 (x) − Qh,m+1 (x)Qh,n (x)

(−1)n r1∗ (x)r2∗ (x) r1m−n (x) − r2m−n (x) = . r1 (x) − r2 (x)

Theorem 3.7. For m any integer number and n any natural number, the generating function of the sequence {Qh, m + n (x)} is as follows

Acknowledgments

∞  n=0

− 2)

2 Qh,n+1 (x)Qh,n−1 (x) − Qh,n (x) = (−1)n r1∗ (x)r2∗ (x).

3

and by the identity (3), the result follows.

r1r (x) r2r (x)

Theorem 3.9 (Cassini’s identity). For any natural number n, we have

n=0

l l=0 r1 (x)el

(−1)n r1∗ (x)r2∗ (x)(r1r (x) − r2r (x))2 . (r1 (x) − r2 (x))2

2 Qh,n+r (x)Qh,n−r (x) − Qh,n (x)

1 gQ (t) = r1 (x) − r2 (x)  3 ∞   (Fh,l+1 (x) − r2 (x)Fh,l (x))el r1n (x)tn 3 

=

Proof. Using the Binet-style formula of Theorem 3.6 and some identities of (9) and (10) involving the roots r1 (x) and r2 (x), we have

.

n=0

l=0

As we stated before, the use of the Binet-style formula can be useful for stating some basic identities. It is the case of Catalan’s identity stated in the following result:

Qh,m (x) + Qh,m−1 (x)t Qh,m+n (x)t = . 1 − h(x)t − t2 n

Proof. Using the Binet-style formula for Qh, n (x) given by (11), we write  ∞ ∞  ∗   r1 (x)r1m+n (x) − r2∗ (x)r2m+n (x) n Qh,m+n (x)tn = t r1 (x) − r2 (x) n=0 n=0   ∞ ∞   1 = r1n (x) − r2∗ (x)r2m (x) r2n (x) tn r1∗ (x)r1m (x) r1 (x) − r2 (x) n=0 n=0   1 1 1 ∗ m ∗ m − r2 (x)r2 (x) = r1 (x)r1 (x) r1 (x) − r2 (x) 1 − r1 (x)t 1 − r2 (x)t and, doing some calculations and the use of Theorem 3.6, we obtain our claim. 

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