Received December 2016. Volume 9, Number 4, 2016 THE BIVARIATE (COMPLEX) FIBONACCI AND LUCAS POLYNOMIALS: AN HISTORICAL INVESTIGATION WITH THE MAPLE´S HELP Francisco Regis Vieira Alves, Paula Maria Machado Cruz Catarino Abstract: The current research around the Fibonacci´s and Lucas´ sequence evidences the scientific vigor of both mathematical models that continue to inspire and provide numerous specializations and generalizations, especially from the sixthies. One of the current of research and investigations around the Generalized Sequence of Lucas, involves it´s polinomial representations. Therefore, with the introduction of one or two variables, we begin to discuss the family of the Bivariate Lucas Polynomias (BLP) and the Bivariate Fibonacci Polynomials (BFP). On the other hand, since it´s representation requires enormous employment of a large algebraic notational system, we explore some particular properties in order to convince the reader about an inductive reasoning that produces a meaning and produces an environment of scientific and historical investigation supported by the technology. Finally, throughout the work we bring several figures that represent some examples of commands and algebraic operations with the CAS Maple that allow to compare properties of the Lucas´polynomials, taking as a reference the classic of Fibonacci´s model that still serves as inspiration for several current studies in Mathematics. Key words: Lucas Sequence, Fibonacci´s polynomials, Historical investigation, CAS Maple. 1. Introduction Undoubtedly, the Fibonacci sequence preserves a character of interest and, at the same time, of mystery, around the numerical properties of a sequence that is originated from a problem related to the infinite reproduction of pairs of rabbits. On the other hand, in several books of History of Mathematics in Brazil and in the other countries (Eves, 1969; Gullberg, 1997; Herz, 1998; Huntley, 1970; Vajda, 1989), we appreciate a naive that usually emphasizes eminently basic and trivial properties related to this sequence. This type of approach can provide a narrow and incongruent understanding of the Fibonacci sequence, mainly about it´s current evolutionary stage. On the other hand, from the work of the mathematicians François Édouard Anatole Lucas (1842 – 1891) and Gabriel Lamé (1795 – 1870), we observe a progressive return by the interest of the study of numerical sequences and their properties. Thus, we highlight the following set of the numerical sequences: 1,1, 2,3,5,8,13, 21,34,55,89,144, 233,377, 610,987,1597, 2584, 4181, 6765,10946, , , n f ; 1,3, 4, 7,11,18, 29, 47, 76,123,199,322,521,843,1364, 2.207,3571,5778,9349,15127, ,L , n ; 1, 2, 5,12, 29, 70,168, 408, , ,P, n ; 1 1 n n n f f f ; 1 1 n n n L L L ; 1 1 2 2 n n n P P L , 1 n . We observed that the initial values are indicated by 1 2 1 2 1 2 1, 1, L ,L 3, P 1, P 2 f f . The Fibonacci polynomials were first studied in 1883 by Belgian mathematician Eugene Charles Catalan (1814 - 1894) and German mathematician Ernest Erich Jacobsthal (1881 - 1965). Thus, Catalan defined the following family of Fibonacci polynomial functions as follows. Definition 1: We will call the Fibonacci Polynomial Sequence - SPF, the set of polynomial functions described by the recurrence relation 1 2 1 2 () 1, () , () () () n n n f x f x xf x xf x f x , 1 n . Undoubtedly, through the previous definition, we can perceive the generalization process that initially occurred with the Fibonacci sequence and, gradually, after a few decades, began to be registered in
25
Embed
THE BIVARIATE (COMPLEX FIBONACCI AND LUCAS ...The Bivariate (Complex) Fibonacci and Lucas Polynomials: an historical investigation with the Maple´s help 73 Volume 9 Number 4, 2016
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Received December 2016.
Volume 9, Number 4, 2016
THE BIVARIATE (COMPLEX) FIBONACCI AND LUCAS
POLYNOMIALS: AN HISTORICAL INVESTIGATION WITH THE
MAPLE´S HELP
Francisco Regis Vieira Alves, Paula Maria Machado Cruz Catarino
Abstract: The current research around the Fibonacci´s and Lucas´ sequence evidences the
scientific vigor of both mathematical models that continue to inspire and provide numerous
specializations and generalizations, especially from the sixthies. One of the current of research and
investigations around the Generalized Sequence of Lucas, involves it´s polinomial representations.
Therefore, with the introduction of one or two variables, we begin to discuss the family of the
Bivariate Lucas Polynomias (BLP) and the Bivariate Fibonacci Polynomials (BFP). On the other
hand, since it´s representation requires enormous employment of a large algebraic notational
system, we explore some particular properties in order to convince the reader about an inductive
reasoning that produces a meaning and produces an environment of scientific and historical
investigation supported by the technology. Finally, throughout the work we bring several figures
that represent some examples of commands and algebraic operations with the CAS Maple that
allow to compare properties of the Lucas´polynomials, taking as a reference the classic of
Fibonacci´s model that still serves as inspiration for several current studies in Mathematics.
Key words: Lucas Sequence, Fibonacci´s polynomials, Historical investigation, CAS Maple.
1. Introduction
Undoubtedly, the Fibonacci sequence preserves a character of interest and, at the same time, of
mystery, around the numerical properties of a sequence that is originated from a problem related to the
infinite reproduction of pairs of rabbits. On the other hand, in several books of History of Mathematics
in Brazil and in the other countries (Eves, 1969; Gullberg, 1997; Herz, 1998; Huntley, 1970; Vajda,
1989), we appreciate a naive that usually emphasizes eminently basic and trivial properties related to
this sequence. This type of approach can provide a narrow and incongruent understanding of the
Fibonacci sequence, mainly about it´s current evolutionary stage.
On the other hand, from the work of the mathematicians François Édouard Anatole Lucas (1842 –
1891) and Gabriel Lamé (1795 – 1870), we observe a progressive return by the interest of the study of
numerical sequences and their properties. Thus, we highlight the following set of the numerical
Parberry, 1969). In particular, we note the results discussed Hoggatt & Long (1974). In fact, we
enunciate the following results, in accordance the formal definition
0 1 1 1( , ) 0,F ( , ) 1, ( , ) ( , ) ( , ), 1n n nF x y x y F x y x F x y y F x y n . With origin in these works
of the seventies, we will announce some important results.
The Bivariate (Complex) Fibonacci and Lucas Polynomials: an historical investigation with the Maple´s help 77
Volume 9 Number 4, 2016
Theorem 2: For 0, 0m n we have 1 1 1( , ) ( , ) ( , ) y ( , ) ( , )m n m n m nF x y F x y F x y F x y F x y .
(Hoggatt & Long, 1974, p. 114).
Proof. We observe that 2 0 1 1 1 0 0 1 1 1 0 1( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )F x y F x y x F x y y F x y F x y F x y y F x y F x y .
Moreover, for 2n we write 3 1 1 1 2 1( , ) ( , ) ( , ) ( , ) 1 1F x y F x y x F x y y F x y x x y
2 2 1 1 1 1 1 1 1 1( , ) ( , ) y ( , ) ( , ) ( , ) ( , ) y ( , ) ( , )F x y F x y F x y F x y F x y F x y F x y F x y . For a fixed
integer 0m we proceed for induction for ‘n’ and we assume that the propertie is true for
1 1 1( , ) ( , ) ( , ) y ( , ) ( , )m n m n m nF x y F x y F x y F x y F x y . In the next step, we will consider the
element 2 1 1 1( , ) ( , ) ( , ) ( ( , ) ( , ) y ( , ) ( , )) ( , )m n m n m n m n m n m nF x y x F x y y F x y x F x y F x y F x y F x y y F x y .
But, we observe that ( 1) 1 1 1( , ) ( , ) ( , ) ( , ) y ( , ) ( , )m n m n m n m nF x y F x y F x y F x y F x y F x y .
Now, we take 2
1 1 1 1( , ) ( , ) y ( , ) ( , ) ( , ) ( , ) y ( , ) ( , )m n m n m n m nxF x y F x y x F x y F x y y F x y F x y F x y F x y
2
1 1 1 1[ ( , ) ( , ) ( , ) ( , )] [ y ( , ) ( , ) y ( , ) ( , )]m n m n m n m nxF x y F x y y F x y F x y x F x y F x y F x y F x y
1 1 1 1 ( 1) 1( , )[ ( , ) ( , )] y ( , )[ ( , ) y ( , )] ( , ) ( , )m n n m n n m nF x y x F x y yF x y F x y xF x y F x y F x y F x y
1y ( , ) ( , )m nF x y F x y . Finally, we obtained that: 2 1 ( 1) 1 1( , ) ( , ) ( , ) y ( , ) ( , )m n m n m nF x y F x y F x y F x y F x y ,
for every 0, 0m n .
Lemma: For 0n , then gcd( , ( , )) 1ny F x y . (Hoggatt & Long, 1974, p. 114)
Proof. The assertion is clearly true for 1n 1 gcd( , ( , )) gcd( ,1) 1y F x y y . Assume that it is true
for any fixed 1k . Then, since 1 1( , ) ( , ) ( , )k k kF x y x F x y y F x y . If the condition does not
hold, we could take 1gcd( , ( , )) ( , )ky F x y d x y . But, en virtue the definion, we have
1( , ) \ ( , ),d(x, y) \ ykd x y F x y . Therefore, we will have 1 1d(x, y) \ ( , ) ( , ) ( , )k k kF x y y F x y x F x y .
However, it can not be d(x, y) \ ( , )kF x y , since gcd( , ( , )) 1ky F x y . Therefore, we obtain that
d(x, y) \ x but, this can not occur unless that 1gcd( , ( , )) 1ky F x y .
Theorem 3: For 0n , then 1gcd( ( , ), ( , )) 1n nF x y F x y . (Hoggatt & Long, 1974, p. 116)
Proof. Again, the result is trivially true for 0, 1n n since that 0 1gcd( ( , ), ( , )) gcd(0,1)F x y F x y
and 1 2gcd( ( , ), ( , )) gcd(1,x) 1F x y F x y . We assume that is true for 1n k , where ‘k’ is a fixed
interger 2k and we assume that 1gcd( ( , ), ( , )) ( , )k kF x y F x y d x y . Since we know
1 1( , ) ( , ) ( , )k k kF x y x F x y y F x y . Again, by definition of the g.c.d., we will have the propertie
1 1( , ) \ ( , ) ( , ) ( , )k k kd x y F x y x F x y y F x y . Thus, we will have 1( , ) \ ( , )kd x y y F x y . But, we
know that 1gcd( ( , ), ( , )) 1k kF x y F x y and, in this way, 1( , )kF x y is not divisible by d( , )x y .
Consequently, ( , ) \d x y y that is a irreductible polynomial in the variable ‘y’. So, it´s a contradiction.
Theorem 4: For 2m , then ( , ) \ ( , ) m\ nm nF x y F x y . (Hoggatt & Long, 1974, p. 116)
Proof. Again, we proceed by induction. Preliminarily, we observe that ( , ) \ ( , )m k mF x y F x y is
true for 11 ( , ) \ ( , )m mk F x y F x y . We proceed by induction, for a fixed integer 1k , that is, we
know that ( , ) \ ( , )m k mF x y F x y , for 1k . In the next step, we will see if the following division
occurs ( 1)( , ) \ ( , )m k mF x y F x y . On the other hand, en virtue the identity of the
78 Francisco Regis Vieira Alves, Paula Maria Machado Cruz Catarino
Acta Didactica Napocensia, ISSN 2065-1430
theorem 1 1 1( , ) ( , ) ( , ) y ( , ) ( , )m n m n m nF x y F x y F x y F x y F x y , we write ( 1) ( , ) ( , )k m km mF x y F x y
1 1( , ) ( , ) y ( , ) ( , )km m km mF x y F x y F x y F x y . But, since we have (by hypothesis) ( , ) \ ( , )m k mF x y F x y and
( , ) \ ( , )m mF x y F x y immediately we obtain 1 1( , ) \ ( , ) ( , ) y ( , ) ( , )m km m km mF x y F x y F x y F x y F x y .
Finally, we find that ( 1)( , ) \ ( , )m k mF x y F x y . This propertie clearly implies that ( , ) \ ( , )m k mF x y F x y ,
for 1k . Thus, we still have \ ( , ) \ ( , )m nm n F x y F x y . Now, for 2m we suppose
( , ) \ ( , )m nF x y F x y and we must obtain that \ nm . On the other hand, if we assume that ‘n’ is not
divisible by ‘m’, by means of the division algorithm, exist intergers ‘q’ and ‘r’ with the condition
,0n m q r r m . Again, by the previous theorem, we take the
formula ( 1) 1 1 1( , ) ( , ) ( , ) ( , ) ( , ) y ( , ) ( , )n m q r m q r m q r m q rF x y F x y F x y F x y F x y F x y F x y .
Finally, we observe that ( , ) \ ( , )m m qF x y F x y by the first part of the proof and ( , ) \ ( , )m nF x y F x y .
Consequently, we still have 1 1( , ) \ ( , ) y ( , ) ( , ) ( , ) ( , )m n m q r m q rF x y F x y F x y F x y F x y F x y .
But, since we know that 1gcd( ( , ), ( , )) 1m q m qF x y F x y and, the only possibility is
( , ) \ ( , )m rF x y F x y but it´s impossible, since we have the condition 0 r m , that is, the term
( , )rF x y is a lower degree than ( , )mF x y . Therefore, 0r and \m n and the proof is complete.
Theorem 5: For 0, 0m n we have gcd( , )gcd( ( , ), ( , )) ( , )m n m nF x y F x y F x y . (Hoggatt & Long,
1974, p. 116)
Proof. Through elementary properties, we know that exist integers r and s, say, 0r and 0s such
that gcd( , ) ( , )m n r m s n r m m n s n . Thus, by theorem, we write ( , )( , ) ( , )r m m n s nF x y F x y
( , ) ( s) ( , ) ( s) 1 ( , ) 1 ( s) ( , ) 1( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )m n n m n n m n n m n s nF x y F x y F x y y F x y F x y F x y F x y
( , ) 1( , ) ( , )m n s ny F x y F x y . But, let d(x, y) gcd( ( , ), ( , ))m nF x y F x y and, consequently we have
d(x, y) \ ( , )r mF x y and d(x, y) \ ( , )s nF x y . From these properties, we still obtain that
( , ) 1 ( , ) 1d(x, y) \ ( , ) ( , ) ( , ) ( , ) ( , )r m m n s n m n s nF x y y F x y F x y F x y F x y . Now, if we conclude that
( , )d(x, y) \ ( , )m nF x y the proof is complete. However, if occur 1d(x, y) \ ( , )s nF x y , we must observe that
11 gcd(d(x, y), ( , ))snF x y . Otherwise, we would have 1'( , ) gcd(d(x, y), ( , ))snd x y F x y and,
we still know 11 gcd( ( , ), ( , ))sn snF x y F x y and d(x, y) \ ( , )s nF x y . Consequently, we can get
that d'(x, y) \ ( , )s nF x y and 1d'(x, y) \ ( , )s nF x y and we must obligatorily have to
11 gcd(d(x, y), ( , ))snF x y . Finally, the property ( , )d(x, y) \ ( , )m nF x y is verified.
We will give some examples below in order to verify the expected behavior of some particular cases,
according to the theorems we have just demonstrated in detail.
18 16 14 2 12 3 10 4 8 5 6 6 4 7
2 8
19
9
17 120 455 1001 1287 92( , )
(is irreductible over Z[x,y],p=19)
4 330
45
F x x x y x y x y x y x y x y x y
y
y
x y
22 20 18 2 16 3 14 4 5 10 6 8 7
6 8 4 9 2 1 1
12
2
0 1
3 21 190 969 3060 6188 8008 6435
3003 71
( , )
(is irreductible over Z[x,y],5 6 =2 )6 p 3
x x y x y x y x y x y x y x y
x y x y x
x
y
F y
y
30 28 2 2 24 3 22 4 20 5 18 6
16 7 14 8 12 9 10 10 8 11 6 12
4 13 2 1 1
31
4 5
29 378 6 2925 14950 53130 134596
245157 319770 293930 184756 75582 18564
2380 120
( , )
(is irreductible over Z[x,y],p=31)
x x y x y x y x y x y x y
x y x y x y x y x y x y
x y x y y
F x y
The Bivariate (Complex) Fibonacci and Lucas Polynomials: an historical investigation with the Maple´s help 79
Volume 9 Number 4, 2016
Before finalizing the list of some important results, we will enunciate a last theorem that provides an
important characterization only for the elements present in the Polynomial Fibonacci´s sequence.
Theorem 6: Let ( , )r r x y be any polynomial in the variables ‘x’ and ‘y’. If there exists a least
positive interger ‘m’, such that ( , ) \ ( , y)mr x y F x , then ( , ) \ ( , y) m\ nnr x y F x . (Hoggatt & Long,
1974, p. 117).
Proof. By the theorem 4, we know if m\ n ( , ) \ ( , )m nF x y F x y . So, we admit that exists a least
positive interger ‘m’, such that ( , ) \ ( , y)mr x y F x and, by transitivity, we conclude ( , ) \ ( , y)nr x y F x .
Now, we suppose that ( , ) \ ( , y)nr x y F x and yet ‘n’ is not divisible by ‘m’. Then, by the Euclidean
Algorithim, exist integers , ,0q s s m and n m q s . Again, by the theorem 2, we can write
( 1) 1 1 1( , y) ( , y) ( , y) ( , y) ( , y) ( , y) ( , y)n m q s m q s m q s m q sF x F x F x F x F x y F x F x . But, since
( , ) \ ( , y)mr x y F x , ( , ) \ ( , y)m qr x y F x and ( , ) \ ( , y)nr x y F x . Consequently, by the last identity, we still
have 1 1( , ) \ ( , y) ( , y) ( , y) ( , y) ( , y)n m q s m q sr x y F x y F x F x F x F x . From this, follows that
1( , ) \ ( , y) ( , y)m q sr x y F x F x and we know 1gcd( ( , y), ( , y)) 1m q m qF x F x . Thus, the only alternative
is ( , ) \ ( , y)sr x y F x , however is a crontradicion, since exists a least positive interger ‘m’, such that
( , ) \ ( , y)mr x y F x and 0 s m . So, ‘n’ is divisible by ‘m’ and the proof is complete.
Now, we will study some properties of the divisibility and factorization of the polynomial elements
present in the Lucas sequence. We will see that it does not enjoy a similar algebraic behavior. Before,
however, let us look at some properties of the matrices.
From the definition 1 2L ( , ) ( , ) ( , ), 1n n nx y x L x y y L x y n we can determine some
particular initial forms: 2
0 1 2L ( , ) 2,L ( , ) ,L ( , ) 2x y x y x x y x y , 3
3L ( , ) 3x y x xy ,
4 2 2
4L ( , ) 4 2x y x x y y , 5 3 2
5L ( , ) 5 5x y x x y xy , ,..etc. Catalani (2002) define the matrix
1( , )
0
xA x y A
y
and 2 2
( , )2
x y yB x y B
xy y
. Immediately, we will have AB
232
2
1 2 2
0 2 (
3
2 )
x xyx x y y x y
y xy y x y y x y
, 2
3
4 2 2 3
2
4 2 3
( 3 ) ( 2 )
x x y y x xy
x xA
xy yB
y y
, 3AB
4 2 3
5 3 2 4 2 2
2 2
5 5 4 2
( 2 2 2 ( 3 ))
x x y xy x x y y
x x y x y y x xyy y
,
6 4 2 2 3 5
4
5 3 2 4
3 2
3 2 2 2 2
6 9 2 5 5
( 2 3 4 ( 2 2 2) )
x x y x y y x x y xy
xAB
yx y x y xy xy x x y x y y y
.
In the figure below, we can analise and conjecture a closed form for the product , 1nAB n . So, from
the appreciation of some particular products, we can acquire a better understanding about the follow
identity 2 1
1
n nn
n n
L LAB
L y L y
, 1n . And, we can still work with the inverse matrix.
On the other hand, we can still get the inverse 1 1
10
( , )
1
yA x y A
x
y
and 1( , )B x y
80 Francisco Regis Vieira Alves, Paula Maria Machado Cruz Catarino
Acta Didactica Napocensia, ISSN 2065-1430
2 2
2
1
2
2
4 4
4 4
2
2
x
Bxy
x y x y
x
x
y x y
y
. From this, we can explore the algebraic behavior of the following
produts: 1 2 3 4 5 6( ) ,( ) ,( ) ,( ) ,( ) ,( ) , ,( ) ,n 1.nAB AB AB AB AB AB AB Moreover, with
the CAS, we can find
2
1 2
2 2 22 2 2 2
2 2 22 2 2 2
1 1 1
2 332
L ( , ) L ( , )2
)( )
L ( , )L ( , )2 3
) )
( 4 ) y ( 4 ) y4 ( 4
( 4 ) y ( 4 ) y( 4 ( 4
x
x y x yx x y
yAB B A
x yx yx y x xy
y x yx y y x y y
x y x yy yx y y x y
,
2
1 2
2 2 22 2 2 2
2 2 22 2 2 2
2 2 2
2 332
L ( , ) L ( , )2
)( )
L ( , )L ( , )2 3
) )
( 4 ) y ( 4 ) y4 ( 4
( 4 ) y ( 4 ) y( 4 ( 4
x
x y x yx x y
yAB B A
x yx yx y x xy
y x yx y y x y y
x y x yy yx y y x y
.
In Figure 4, we can observe the behavior of the product of the matrices indicated earlier and, through
some preliminary cases, by means of an inductive process, formulate it´s general term. Of course, we
can understand that calculus becomes impractical without the use of technology.
Now let us look at some properties of divisibility and factorization of some of the polynomial
functions into two variables, present in the Bivariate Lucas Sequence. In Figure 5, we considered a
strong propertie that permits determine any term of the BPL. In the left side, we can see fourth order
matrix and, in the right side, we have considered a fifth order matrix. We calculate that
4 3 1
3 2 2det( ( , )) ( 3 ( 3 ) (, ) ( , ) 3 )x xyH x y L x y L x yx x y x y and 4 2
5
2
4det( ( , )) ( , 2) 4H x y L x x x y yy .
We observed that the polynomial in two variables 4 2 24 2x x y y is irreductible, while we found
2
3 1 ( 3 )( , ) ( , )L x y L x x yy . Moreover, we have 8 6 4 2 2 3 4
9 8det( ( , )) ( , ) 8 20 16 2x x y x y x y yH x y L x y
is a another irreductible polynomial over the field [x, y]IR . On the other hand, in Figure 5, we
observe 12 10 8 2 6 3 4 4 2 5
1
6
3 12 12 54 112 10det( ( , )) ( , 5 36) 2x x y x y x y xH x y L x y y x y y and, by a
command of the CAS, we can write 4 2 2 8 6 4 2 2 3
13 12
4( 4det( ( , )) ( , ) )2 ( 8 20 16 )x x y x xH x y L x y x y x y yy y
4 8( , ) ( , )L x y L x y , however 12 ( , )L x y is not divisible by 2 ( , )L x y and 12 is not divisible by 8.
The Bivariate (Complex) Fibonacci and Lucas Polynomials: an historical investigation with the Maple´s help 81
Volume 9 Number 4, 2016
Figura 4. With the CAS Maple we can verify the algebraic behavior of the product of the matrices
In addition, with the use of software, we can also determine the factorization and, therefore, the
decomposition of irreductible factors of the polynomial terms over the ring Z[x,y] . We observe in the
list below that, unlike the case of Fibonacci, we will have elements of prime index that admit a
factorization in irreducible factors ( 5( , )L x y , 11( , )L x y , 17 ( , )L x y ).
5 3 2 4 2 2
6 5det( ( , )) ( , ) (reductible over 5 5 ( 5 5 ) Z[x,y],p=5)x x y xy x xH x y L x y x y y
8 6
12 11
11 9 7 2 5 3 3 4 5 10 2
3 4 5
1
4 2
det( ( , )) ( , ) ( , )(11 44 77 55 11 11 44
77 55 11 ) (reductible over Z[x,y],p=11)
x x y x y x y x y xy x x y x y
x
H x y L x y L x y
y x y y
13 12
4 2 2 8 6 4 2 2 3 4 8 6 4 2
2
4
4
3 4
12
det( ( , )) ( , ) ) ( , )( 4 2 ( 8 20 16 ) ( 8 20
16 ) is red( , ) \ ( , ), over Z[x,y]uctible
H x y L x x x y x x y x y x y y xy y L x y
L x y L
x y x y
x yy xy
14 12 10 2 8 3 6 4 4 5 2 6 7
2 12 10 8 2 6 3 4 4 2
15 14
5 6 12 10 8 2
6 3 4 4 2 5 6
2
14 77 210 294 196 49 2
( 2 )( 12 53 104 86 24 ) ( 12 53
104 86 24 ) is reduc
det( ( , )) ( ,
tibl
)
( ,
e
)
over
x x y x y x y x y x y x y y
x y x x y x y x y x y x y y x x y x y
x y x y x y
H x y L x
y
y
L x y
Z[x,y]
82 Francisco Regis Vieira Alves, Paula Maria Machado Cruz Catarino
Acta Didactica Napocensia, ISSN 2065-1430
15 13 11 2 9 3 7 4 5 5 3 6 7
2 4 2 2 8 6 4 2 2 3 4 2 4 2 2
8 6 4 2 2
16 15
3
1
4
15 90 275 450 378 140 15
( 3 )( 5 5 )( 7 14 8 ) ( 3 )( 5 5 )
( 7 14 8 ) is re
det( ( , )) ( , )
( , )
over ductible
H x y L x y
L x y
x x y x y x y x y x y x y xy
x x y x x y y x x y x y x y y x y x x y y
x x y x y x y y
Z[x,y]
16 14 12 2 10 3 8 4 6 5 4 6
2 7
17 16
8
det( ( , )) 16 104 352 660 672 336( , )
(is irreductible over Z[x,y])64 2
x x y x y x y x y x y x yH x y L x y
x y y
17 15 13 2 11 3 9 4 7 5 5 6
3 7 8 16 14 12 2 10 3 8 4 6 5 4
18 1
7
7
6 2
8
17 119 442 935 1122 714
204 17 ( 17 119 442 935 1
det( ( , )) ( , )
(reductible over
122 714 204
17 Z) [x,y],p=17)
x x y x y x y x y x y x y
x y xy x x x y x y x y x y x
H x y L x
y x y x
y
y
y
18 16 14 2 12 3 10 4 8 5
6 6 4 7 2 8 9 2 4 2 2 12 10 8 2
6 3 4 4 2 5 6 4 2 2 12 10 8
19 18
2
18 135 546 1287 1782
1386
det( ( , )) ( , )
(
540 81 2 ( 2 )( 4 )( 12 54
112 105 36 ) ( 4 )( 12 5, 4)
x x y x y x y x y x y
x y x y x y y x
H x y L x y
L x y
y x x y y x x y x y
x y x y x y y x x y y x x y x y
2 6 3
4 4 2 5 6
112
105 36 ) is reductible over Z[x,y]
x y
x y x y y
19 17 15 2 13 3 11 4 9 5 7 6
5 7 3 8 9 18 16 14 2 12 3 10 4 9 5
6 7 8 9 1
20 9
8
1
6 4 2
1
19 152 665 1729 2717 25det( ( , )) ( , )
(
)
08
1254 285 19 19 152 665 1729 2717
2508 1254 285 ( , )1 (9 1
x x y x y x y x y x y x y
x y x y xy
H x y L x y
x
L x y
x x y x y x y x y x y
x y x y x y y x
6 4 2
1 19
16 14 2 12 3 10 4
9 5 6 7 8 9
9 152 665 1729
2717 2508 1254 285 19 is re) ( , ) \ ( , ) over Z[x,y]ductible
x y x y x y x y
x y x y x y x y L x y L xy y
Now, from the list of decomposition into irreducible factors of the elements of the bivariate
polynomial sequence, we can understand that: 1 3( , ) \ ( , )L x y L x y and 3( , )L x y is not divisible by
2
2( , ) 2L y xx y . 8 6 4 2 2 3 4
8 8 2( , 1 2) 0 6L x y x x y x y x y y is a irreductible polynomial and
is not divisible by 2
2( , ) 2L y xx y or 4
4
2 2( 2, ) 4L x x yx yy . In the same manner, the
bivariate polynomial12 10 8 2 6 3 4
1
6
2
4 2 512 54 112 105 36 2( , ) x x y x yL x y x x y yx y y is reducitible
over the ring [x, y]IR , since we obtained 8
13 12 4
6 4 2 2 3 4det( ( , )) ( , () 8 20 16 )( , )H x y L x y L x y x x y x y x y y ,
that is 4 12( , ) \ ( , )L x y L x y , while 12 ( , )L x y is not divisible by 3( , )L x y or 6 ( , )L x y . On the other
hand, we observe 3 15( , ) \ ( , )L x y L x y and, however, 15( , )L x y is not divisible by 5( , )L x y .
Moreover, we can conclude that 16 ( , )L x y is not divisible by the following
elements 2 ( , )L x y , 4 ( , )L x y and 8( , )L x y . And, the only division propertie is 1 16( , ) \ ( , )L x y L x y .
According to the result indicated by the software, 17 ( , )L x y is irreducitible polynomial over the ring
[x, y]IR . In addition, we further determined that 19 ( , )L x y , that despite having a prime subscript,
have the element 3( , )L x y as a irreducitible factor. Finally, we conclude that 20 ( , )L x y is not
divisible by 2 ( , )L x y , 5( , )L x y or 10 ( , )L x y . In Figure 5, we visualize some command employed en
virtue to determine it´s decomposition over the ring [x, y]Z .
Now, in addition to an extensive set of algebraic expressions provided by software that indicate the
decomposition of polynomial functions into two variables, we can draw some conclusions regarding
The Bivariate (Complex) Fibonacci and Lucas Polynomials: an historical investigation with the Maple´s help 83
Volume 9 Number 4, 2016
the character of divisibility and factorization of Lucas' polynomial functions. For example, in Figure 5,
on the left side, we observe that 8 6 4 2 2 3 4
9 9det( ( , )) ( , ) 8 20 16 2x x y x y x y yH x y L x y is
irreductible, since the software when using the factor command factor[ ], produces the same algebraic
expression. But, in the same figure, on the right side, we visualize that the element
13 12
4 2 2 8 6 4 2 2 3 4 8 6 4
4
2det( ( , )) ( , ) ) ( , )( 4 2 ( 8 20 16 ) ( 8 20x x y x x yH x y x y x y y x x yL y L x y xx yy
3
4 8
2 4 ( , , )) ) (16 L x y yy Lx y x has two components as irreductible polynomials. So, like we have
mentioned, we know 4 12( , ) \ ( , )L x y L x y , however 12 ( , )L x y is not divisible by 2 ( , )L x y .
Figura 5. We obtain the decomposition of the irreducible factors of the elements of the set of polynomials with CAS Maple
In the next section, we will address an explicit formula for the polynomial terms present in both
sequences. Yet, we will enunciate some properties involving the Greatest Common Divisor of
polynomial functions into two variables, now with the introduction of an imaginary unit 2 1i . In
this way we can compare the class of the BFP (definition 4) with the class of BCFP (definition 6). We
will find the regularity and invariance of several properties indicated in the theorems discussed here
and, conversely, the same regularity cannot be observed in the class of BLP (definition 5) and the
BCLP (definition 7). In fact, we showed that ( , ) \ ( , ) m\ nm nF x y F x y . But, with the software, we
found several exemples that are counterexamples.
84 Francisco Regis Vieira Alves, Paula Maria Machado Cruz Catarino
Acta Didactica Napocensia, ISSN 2065-1430
Figura 6. We obtain the decomposition of the irreducible factors of the elements of the set of polynomials with CAS Maple
4. Some properties of the Bivariate Complex Lucas Polynomial Sequence with the
Maple´s help.
In the previous section, we discuss some properties of bivariate polynomials in two real variables. We
now turn to the study of a special class of Bivariate Complex Fibonacci and Lucas Polynomials,
originating from the introduction of an imaginary unit ‘i’ and inherit a tendency of the works
interested in the process of complexification of the Fibonacci model (Iakin, 1977; King, 1968; Scott,
1968; Waddill & Sacks, 1967). With this, we can further discuss the process of complexing said
recursive sequence. Before, however, we recall the definition presented in the predecessor sections en
virtue to present our first theorem.
Theorem 2: For 0n we have ( , ) ( , )
( , )( , ) ( , )
n n
n
x y x yF x y
x y x y
and ( , ) ( , ) ( , )n n
nL x y x y x y .
The Bivariate (Complex) Fibonacci and Lucas Polynomials: an historical investigation with the Maple´s help 85
Volume 9 Number 4, 2016
Proof. We the characteristic equation designated by 2 0t ix t y . Consequently, we will have the
following properties and relations between the roots
24( , )
2
ix y xx y
,
24( , )
2
ix y xx y
,
( , ) ( , )x y x y y . Finally, from the recurrence relation, we write: 1 1( , ) ( , ) ( , )n n nF x y ix F x y y F x y