Determinantal and permanental representations of convolved Lucas polynomials

Applied Mathematics and Computation 281 (2016) 314–322

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Determinantal and permanental representations of convolved Lucas polynomials Adem S¸ ahin a, José L. Ramírez b,∗ a b

Faculty of Education, Gaziosmanpas¸ a University, Tokat 60250, Turkey Departamento de Matemáticas, Universidad Sergio Arboleda, Bogotá 110221, Colombia

a r t i c l e

i n f o

MSC: Primary 11B39 Secondary 11C20 15B36

a b s t r a c t In the present paper, we define the convolved generalized Lucas polynomials as an extension of the classical convolved Fibonacci numbers. Then we give some determinantal and permanental representations of this new integer sequence. © 2016 Elsevier Inc. All rights reserved.

Keywords: Convolved (p, q)-Fibonacci polynomials Determinantal Permanental Hessenberg matrices

1. Introduction The generalized Lucas polynomial or (p, q)-Fibonacci polynomial is a recursive sequence that generalizes several polynomial sequences defined by recurrence relations of order two. Some known examples of (p, q)-Fibonacci polynomials are the Fibonacci numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Morgan–Voyce polynomials, Pell polynomials, Jacobsthal polynomials, Fermat polynomials, among others; see Table 1. Let p(x) and q(x) be polynomials with real coefficients. The (p, q) -Fibonacci polynomials {Fp,q,n (x )}n∈N are defined by the recurrence relation

Fp,q,0 (x ) = 0, Fp,q,1 (x ) = 1, Fp,q,n+1 (x ) = p(x )Fp,q,n (x ) + q(x )Fp,q,n−1 (x ), n  1.

(1)

This sequence was first studied by Lucas in a series of papers [17–19] and then forgotten. Hoggatt and Long [7] rediscovered this sequence. Recently, Lee and Asci [13] obtained factorizations of Pascal matrix involving (p, q)-Fibonacci polynomials by using Riordan arrays. Cheon et al. [2] gave a combinatorial interpretation by using lattice paths. Wang [25] found some arithmetic properties by applying some elementary methods. Amdeberhan et al. [1] studied several combinatorial identities from a combinatorial approach and they introduced a generalized Fibonomial and Lucanomial coefficients in analogy with the definition of binomial coefficients. From Eq. (1) is clear that the generating function of the (p, q)-Fibonacci polynomials is ∞  n=0

Fp,q,n (x )zn =

z . 1 − p( x ) z − q ( x ) z 2

Particular cases of the (p, q)-Fibonacci polynomials are listed in Table 1. ∗

Corresponding author. Tel.: +57 325 7500x2831. E-mail addresses: [email protected] (A. S¸ ahin), [email protected], [email protected] (J.L. Ramírez).

http://dx.doi.org/10.1016/j.amc.2016.01.064 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.

A. S¸ ahin, J.L. Ramírez / Applied Mathematics and Computation 281 (2016) 314–322

315

Table 1 Special cases of the polynomials Fp,q,n (x ). p(x)

q(x)

Fp, q, n (x)

x x 2x 2x 3x x+2

1 −α 1 −1 −2 −1

Fibonacci polynomials Fn (x) Dickson polynomials of the second kind En−1 (x, α ) Pell polynomials Pn (x) Chebyshev polynomials of the second kind Un (x) Fermat polynomials Fn (x ) Morgan–Voyce polynomials Bn (x)

From the generating function of the Fibonacci sequence the convolved Fibonacci numbers Fj(r ) can be defined by

(1 − x − x2 )−r =

∞  (r ) j Fj+1 x,

r ∈ Z+ .

(2)

j=0

If r = 1 we have classical Fibonacci numbers. These numbers and their generalizations have been studied in several papers during the last thirty years; see, e.g., [4,6,16,21,23,24,28,29]. In analogy to Eq. (2) we introduce the convolved (p, q)-Fibonacci polynomials. Specifically, the convolved (p, q)-Fibonacci (r ) polynomials or convolved generalized Lucas polynomials, say Fp,q,n (x ), are defined by r) g(p,q (t ) := (1 − p(x )t − q(x )t 2 )−r =

∞  (r ) Fp,q,n (x )t n , r ∈ Z+ . +1

(3)

n=0

Then we derive some basic combinatorial identities such as recurrence relations. After that, we obtain some determinantal and permanental representations by using various Hessenberg matrices. Similar researches have been made for similar combinatorial sequences. For example, Minc [20] defined an n × n (0,1)matrix F(n, k), and showed that the permanents of F(n, k) are equal to the generalized order-k Fibonacci numbers. Yilmaz and Bozkurt [27] obtained some relations between Padovan sequence and permanents of one type of Hessenberg matrix. Kiliç and Stakhov [12] gave permanent representation of Fibonacci and Lucas p-numbers. Öcal et al. [22] studied some determinantal and permanental representations of k -generalized Fibonacci and Lucas numbers. Janjic´ [8] considered a particular upper Hessenberg matrix and showed its relations with a generalization of the Fibonacci numbers. Kaygısız and S¸ ahin [11] gave a method to calculate successive n terms of generalized order-k Fibonacci numbers by using Hessenberg and triangular matrices. Li [15] obtained three new Fibonacci-Hessenberg matrices and studied its relations with Pell and Perrin sequence. More examples can be found in [9,10,14,26]. 2. Convolved (p, q)-Fibonacci polynomials From definition of convolved (p, q)-Fibonacci polynomials we obtain that (r ) Fp,q,n (x ) = +1



Fp,q, j1 +1 (x )Fp,q, j2 +1 (x ) . . . Fp,q, jr +1 (x ).

j1 + j2 +...+ jr =n

Moreover, by the binomial series we get









 

j ∞ ∞    −r −r j g p,q (t ) = (−1 ) j ( p(x )t + q(x )t 2 ) j = (−1 ) j ( p(x )t ) j−k (q(x )t 2 )k j j k

(r )

j=0

=

  ∞  

 (−1 )l−k

l=0 k=0

2 ∞   l=0 k=0



−r l−k



l

=

(−1 )l−k



−r k

l





2 ∞   r+k−1 = k l=0 k=0



l−k p(x )l−2k q(x )k t l k



−r − k p(x )l−2k q(x )k t l l − 2k



r+l−k−1 p(x )l−2k q(x )k t l . l − 2k

Note that in the last equality we use the binomial identity get the following identity (r ) Fp,q,n (x ) = +1

k=0

j=0

l 2

 n/2  j=0



r+ j−1 j

r  k

= (−1 )k

k−r−1 k

. Comparing the coefficients on both sides, we



r + n − j − 1 n−2 j p ( x )q j ( x ). n − 2j

(4)

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A. S¸ ahin, J.L. Ramírez / Applied Mathematics and Computation 281 (2016) 314–322 Table 2 (r ) Fp,q,n (x ), with r = 1, 2, 3. n

(1 ) Fp,q,n (x )

(2 ) Fp,q,n (x )

(3 ) Fp,q,n (x )

0 1 2 3 4 5 6 7 8

0 1 p p2 p3 p4 p5 p6 p7

0 1 2p 3 p2 4 p3 5 p4 6 p5 7 p6 8 p7

0 1 3p 6 p2 + 3q 10 p3 + 12qp 15 p4 + 30qp2 + 6q2 21 p5 + 60qp3 + 30q2 p 28 p6 + 105qp4 + 90q2 p2 + 10q3 36 p7 + 168qp5 + 210q2 p3 + 60q3 p

+q + 2qp + 3qp2 + 4qp3 + 5qp4 + 6qp5

+ q2 + 3q2 p + 6q2 p2 + q3 + 10q2 p3 + 4q3 p

+ 2q + 6qp + 12qp2 + 3q2 + 20qp3 + 12q2 p + 30qp4 + 30q2 p2 + 4q3 + 42qp5 + 60q2 p3 + 20q3 p

In particular, if r = 1 we obtain a combinatorial formula to evaluate the (p, q)-Fibonacci polynomials

Fp,q,n (x ) =

n−1   2 



n − i − 1 n−2i p ( x )qi ( x ). i

i=0

In Table 2 some polynomials of convolved (p, q)-Fibonacci polynomials are provided. Theorem 1. The following identities hold (r ) i. Fp,q, (x ) = r p(x ). 2

(r ) (r ) (r ) (r−1 ) ii. Fp,q,n (x ) = p(x )Fp,q,n (x ) + q(x )Fp,q,n (x ) + Fp,q,n ( x ), n  2. −1 −2

(r ) (r+1 ) (r+1 ) iii. nFp,q,n (x ) = r ( p(x )Fp,q,n (x ) + 2q(x )Fp,q,n (x )), n  1. +1 −1

Proof. i. Taking n = 1 in (4), we obtain (r ) Fp,q, (x ) = 2



 

r−1 0

r p( x ) = r p ( x ) . 1

ii. The equality follows from observing that ∞ ∞ ∞    (r ) (r ) (r−1 ) n 2 n Fp,q,n ( x ) t = ( p ( x ) t + q ( x ) t ) F ( x ) t + Fp,q,n (x )t n . +1 p,q,n+1 +1 n=0

n=0

(r )

iii. Taking the first derivative of g p,q (t ) = (1 − p(x )t r) (g(p,q (t )) =

 ∞  (r ) Fp,q,n (x )nt n−1 = r +1 n=1

n=0

− q(x )t 2 )−r ,

we obtain

1 1 − p(x )t − q(x )t 2

r−1 

p(x ) + 2q(x )t (1 − p(x )t − q(x )t 2 )2

r+1 ) = r ( p(x ) + 2q(x )t )g(p,q (t ).

Therefore the identity is clear.



(5)



Theorem 2. The convolved (p, q)-Fibonacci polynomials satisfy the following linear recurrence (r ) Fp,q,n (x ) =

r+n−2 n + 2r − 3 (r ) (r ) p(x )Fp,q,n (x ) + q(x )Fp,q,n ( x ), n  2, −1 −2 n−1 n−1

(r ) (r ) with the initial values Fp,q, (x ) = 0 and Fp,q, ( x ) = 1. 0 1

Proof. Multiplying both sides of Eq. (5) by (1 − p(x )t − q(x )t 2 ), we get r) r) (1 − p(x )t − q(x )t 2 )(g(p,q (t )) = r ( p(x ) + 2q(x )t )g(p,q (t ).

By comparing the coefficients we have (r ) (r ) (r ) nFp,q,n (x ) − p(x )(n − 1 )Fp,q,n (x ) − q(x )(n − 2 )Fp,q,n (x ) +1 −1 (r ) (r ) = r ( p(x )Fp,q,n (x ) + 2q(x )Fp,q,n (x )). −1

Therefore, Eq. (6) follows.



(6)

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317

3. Computing convolved (p, q)-Fibonacci polynomials by using determinant of some Hessenberg matrices and inverse of triangular matrices An n × n matrix An = (ai j ) is called lower Hessenberg matrix if ai j = 0 when j − i > 1, i.e.,

⎡

a11

⎢ a21 ⎢ ⎢ a ⎢ 31 An = ⎢ ⎢ .. ⎢ . ⎢ ⎣an−1,1 an,1

a12

0

···

a22

a23

···

a32

a33

···

.. .

.. .

an−1,2

an−1,3

···

an,2

an,3

···

0

⎤

⎥ ⎥ 0 ⎥ ⎥ ⎥ .. ⎥. . ⎥ ⎥ an−1,n ⎦ 0

an,n

The (p, q)-Fibonacci polynomials can be obtained by taking the determinant of an n × n Toeplitz–Hessenberg matrix. Specifically,

⎡

p( x )

⎢ q (x ) ⎢ ⎢ 0 ⎢ Fp,q,n+1 (x ) = det ⎢ ⎢ .. ⎢ . ⎢ ⎣ 0

−1

0

···

p( x )

−1

···

q (x )

p( x )

···

.. .

.. .

0

0

···

0

0

···

0

0

⎤

⎥ ⎥ 0 ⎥ ⎥ ⎥ .. ⎥ . . ⎥ ⎥ −1 ⎦ p(x ) n×n 0

(7)

It is clear by the Laplace expansion of the first row and the recurrence relation (1). (r ) (r ) Theorem 3. Let n ≥ 1 be an integer, Fp,q,n (x ) be the nth convolved (p, q)-Fibonacci polynomial and K p,q,n = (kst ) be an n × n Hessenberg matrix, defined as

⎧r + s − 1 ⎪ p( x ) , ⎪ ⎪ s ⎪ ⎪ ⎨ 2r + s − 2 q ( x ), kst = s ⎪ ⎪ ⎪−1, ⎪ ⎪ ⎩ 0,

if t = s; if s − t = 1; if t − s = 1; otherwise.

Then (r ) (r ) det(K p,q,n ) = Fp,q,n ( x ). +1

Proof. We proceed by induction on n. The result clearly holds for n = 1. Now suppose that the result is true for all posi(r ) (r ) (r ) (r ) tive integers less than or equal to n − 1. We prove it for n. From Eq. (6), we get K p,q,n · [Fp,q, (x ), Fp,q, (x ), . . . , Fp,q,n ( x )] T = 1 2 (r ) [0, 0, . . . , 0, Fp,q,n (x )]T . By Cramer’s rule we get +1 (r ) Fp,q,n (x ) =

(r ) det(K p,q,n )F ( r ) ( x ) −1 p,q,n+1 (r ) det(K p,q,n )

Then (r ) Fp,q,n (x ) = +1

(r ) (r ) det(K p,q,n )Fp,q,n (x ) (r ) det(K p,q,n ) −1

(r ) (r ) From the hypothesis of induction we obtain Fp,q,n (x ) = det(K p,q,n ). +1



Note that the above procedure can be adapted to prove that any linear recurrence relation of order m can be expressed in terms of determinants of sections of the matrix of recurrence coefficients.

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(2 ) Example 4. We obtain the polynomial Fp,q, (x ) by using Theorem 3: 5

⎡

−1

2p

⎢ ⎢4 ⎢ q ⎢2 det ⎢ ⎢ ⎢0 ⎢ ⎣

0

3 p 2 5 q 3

0

0

⎥ ⎥ ⎥ ⎥ ⎥ = 3q2 + 5 p4 + 12 p2 q. ⎥ −1 ⎥ ⎥ ⎦

−1

0

4 p 3 6 q 4

0

⎤

5 p 4

(r ) If we multiply the kth column by (−1 )(−i )k and the jth row by (−1 )i j of the matrix K p,q,n , where i is the imaginary unit, then the determinant is not altered. Therefore we get the following corollary (r ) (r ) Corollary 5. Let n ≥ 1 be an integer, Fp,q,n (x ) be the nth convolved (p, q)-Fibonacci polynomial and H p,q,n = (hst ) be an n × n Hessenberg matrix defined as

⎧r + s − 1 ⎪ p( x ) , ⎪ ⎪ s ⎪ ⎪ ⎨ 2r + s − 2 q(x )i, hst = s ⎪ ⎪ ⎪ ⎪ ⎪ ⎩i, 0,

Where i =

if t = s; if s − t = 1; if t − s = 1; otherwise.

√ −1. Then

(r ) (r ) det(H p,q,n ) = Fp,q,n ( x ). +1

(8)

(3 ) Example 6. We obtain the polynomial Fp,q, (x ) by using Corollary 5: 5

⎡

3p 6 qi 2

⎢ ⎢ ⎢ ⎢ det ⎢ ⎢ ⎢ ⎣

i 4 p 2 7 qi 3

0 0

0

0

0

i

0

5 p 3 8 qi 4

i 6 p 4

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ = 6q2 + 15 p4 + 30 p2 q. ⎥ ⎥ ⎦

The permanent of a matrix is defined in a similar manner to the determinant but all the sign used in the Laplace expansion of minors are positive. The permanent of a n-square matrix is defined by

perA =

n 

σ ∈Sn i=1

a iσ ( i ) ,

where the summation extends over all permutations σ of the symmetric group Sn (cf. [20]). There is a relation between permanent and determinant of a Hessenberg matrix (cf. [5,10]). Let A be a lower Hessenberg matrix, S be an n-square (1, −1 ) matrix defined as si, j = −1 if j = i + 1 and otherwise 1, i.e.,

⎡1

−1

⎢1 ⎢ ⎢ S=⎢ ⎢ ⎣1

1

1

−1

···

1

···

1

1

⎤

⎥ ⎥ ⎥ ⎥. ⎥ −1⎦ 1 .. .

.. .

···

1

···

1

1

···

1

1

Then per A = det(A ◦ S ), where A ◦ S denotes Hadamard product of A and S. Therefore it is clear the following corollary.

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319

(r ) r) (r ) Corollary 7. Let n ≥ 1 be an integer, Fp,q,n (x ) be the nth convolved (p, q)-Fibonacci polynomial, L(p,q,n = (lst ) and J p,q,n = ( jst ) be n × n Hessenberg matrices defined as

⎧r + s − 1 ⎪ p( x ) , ⎪ ⎪ s ⎪ ⎪ ⎨ 2r + s − 2 q ( x ), lst = s ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎩

if t = s; if s − t = 1; if t − s = 1;

0,

and

otherwise,

⎧r + s − 1 ⎪ p( x ) , ⎪ ⎪ s ⎪ ⎪ ⎨ 2r + s − 2 q(x )i, jst = s ⎪ ⎪ ⎪ ⎪ −i, ⎪ ⎩

if t = s; if s − t = 1; if t − s = 1;

0,

where i =

otherwise.

√ −1. Then

r) (r ) (r ) per(L(p,q,n ) = per(Jp,q,n ) = Fp,q,n ( x ). +1

(2 ) Example 8. We obtain the polynomial Fp,q, (x ) by using Corollary 7: 5

⎡

2p ⎢4 ⎢ q ⎢2

1 3 p 2 5 q 3

⎢ ⎢0 ⎢ ⎣

per⎢

0

0

0

⎤

0

⎡

2p ⎢4 ⎢ qi ⎢2

⎥ ⎥ ⎥ ⎥ ⎢ ⎥ = per⎢ ⎢ 0 1 ⎥ ⎥ ⎢ ⎦ ⎣

1

0

4 p 3 6 q 4

5 p 4

−i 3 p 2 5 qi 3

0

0

0 −i 4 p 3 6 qi 4

0

⎤

⎥ ⎥ ⎥ ⎥ ⎥ = 3q2 + 5 p4 + 12 p2 q. −i ⎥ ⎥ ⎦ 0

5 p 4

Chen and Yu [3] presented two nonsingular matrices:

⎡

h11

⎢ ⎢ h ⎢ 21 ⎢ H = ⎢ .. ⎢ . ⎢ ⎣hn−1,1 ⎡

hn,1

1

0

⎢ ⎢ H˜ = ⎢ ⎣

···

h12

0

h22

h23

.. .

..

hn−1,2

···

hn−1,n−1

···

hn,n−1

hn,2 ···

0

.

⎤

..

.

..

.

0

⎤

⎥ ⎥ ⎥ ⎥ ⎥, and 0 ⎥ ⎥ hn−1,n ⎦ .. .

hn,n

.. ⎥ .⎥ ⎥.

0⎦

H

1 They obtained the following properties:



˜ −1

H

=

[α ]n×1 h



[L]n×n

βT





1×n

,

(9)

(n+1 )×(n+1 )

det(H ) = (−1 )n h · det(H˜ ),

(10)

H α + hen = 0,

(11)

where en is nth column of matrix In .

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(r ) Theorem 9. Let n ≥ 1 be an integer, H p,q,n be the n × n Hessenberg matrix defined in Corollary 5 and

⎡1

···

0

0

⎢

⎢ (r ) H˜ p,q,n := ⎢ ⎣

⎤

.. ⎥ . ⎥.

⎥

0⎦

(r ) H p,q,n

1 (r ) −1 Then, the first column of the inverse matrix (H˜ p,q,n ) is

⎡

⎤

1

(r ) ⎢ iFp,q, (x ) ⎥ 2 ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥, . ⎢ ⎥ ⎢ n−1 (r ) ⎥ ⎣ i Fp,q,n (x ) ⎦ (r ) in+1 Fp,q,n (x ) +1

√ (r ) where Fp,q,n (x ) is the nth convolved (p, q)-Fibonacci polynomial and i = −1. Proof. Let us construct the matrix



(r ) −1 (H˜ p,q,n ) =

[α ]n×1 h



[L]n×n

βT





1×n

(n+1 )×(n+1 )

and obtain the entries of the first column. By using (10) and (8) we obtain (r ) (r ) det(H p,q,n ) = (−1 )n h · det(H˜ p,q,n ),

then

h=

(r ) det(H p,q,n ) (r ) (−1 )n det(H˜ p,q,n )

(r ) Fp,q,n (x ) +1

=

(−1 )n in−1

(r ) = in+1 Fp,q,n ( x ). +1

From relation (11) we get

[α ] = − (

(r )

⎡

(r ) H p,q,n −1 in+1 Fp,q,n +1

)

⎤

1

⎢ iF (r ) (x ) ⎥ ⎢ p,q,2 ⎥ ⎥. ( x )en = ⎢ .. ⎢ ⎥ . ⎣ ⎦ (r ) in−1 Fp,q,n (x )

Therefore, we obtain the required result.



(3 ) (x ) for n = 1, 2, 3, 4, 5 by using Theorem 9. Example 10. We obtain Fp,q,n

⎡

(3 ) H p,q, 4

1

⎢ 3p ⎢ ⎢6 ⎢ qi ⎢ = ⎢2 ⎢ ⎢ 0 ⎢ ⎣ 0

(3 ) −1 (H˜ p,q, ) 4

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

0

0

0

i 4 p 2 7 qi 3

0

0

0⎥

i

0

0⎥

5 p 3 8 qi 4

0

⎡

i 6 p 4

1

⎥ ⎥ ⎥ ⎥, and ⎥ 0⎥ ⎥ ⎦

1 0

0

0

0⎥

3ip

−i

0

0

2p 7 10 2 iq + ip 3 3 15 − ipq − 5ip3 2

−i 5 p 3 5 −2q − p2 2

0

6iq + 15ip + 30ip q 4

2

⎤

0

−3q − 6 p2 −12ipq − 10ip3 2

⎤

0

⎥

0⎥ ⎥

⎥

−i

0⎥ ⎥

3 ip 2

1

⎦

A. S¸ ahin, J.L. Ramírez / Applied Mathematics and Computation 281 (2016) 314–322

Then

⎡

⎤

1

⎡

⎢ iFp,q,2 (x ) ⎥ ⎢ ⎢ ⎥ ⎢ −F (3) (x ) ⎥ ⎢ ⎢ p,q,3 ⎥ = ⎢ ⎢ (3 ) ⎥ ⎢ ⎣−iFp,q,4 (x )⎦ ⎣ (r )

⎤

1

(3 )

⎥ ⎥ ⎥. ⎥ ⎦

3ip −3q − 6 p2 −12ipq − 10ip3

6iq + 15ip4 + 30ip2 q 2

iFp,q,5 (x )

(r ) Theorem 11. Let n ≥ 1 be an integer, K p,q,n be the n × n Hessenberg matrix defined in Theorem 3 and

⎡1

···

0

0

⎢

⎢ (r ) K˜ p,q,n := ⎢ ⎣

⎤

.. ⎥ . ⎥.

⎥

0⎦

(r ) K p,q,n

1 (r ) −1 Then, the first column of the inverse matrix (K˜ p,q,n ) is

⎡

⎤

1

⎢ Fp,q,2 (x ) ⎥ ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥, .. ⎢ ⎥ ⎢ (r ) ⎥ ⎣ Fp,q,n (x ) ⎦ (r ) −Fp,q,n (x ) +1 (r )

(r ) where Fp,q,n (x ) is the nth convolved (p, q)-Fibonacci polynomial.



Proof. The proof runs like in Theorem 9. (2 )

Example 12. We obtain Fp,q,n (x ) for n = 1, 2, 3, 4, 5 by using Theorem 11.

⎡

(2 ) K˜ p,q, 4

⎢ 2p ⎢ ⎢4 ⎢ q ⎢ = ⎢2 ⎢ ⎢0 ⎢ ⎣ ⎡

(2 ) −1 (K˜ p,q, ) 4

Then

⎡

1

0

0

0

0

−1 3 p 2 5 q 3

0

0

0⎥

−1

0

0⎥ ⎥ ⎥,

0

4 p 3 6 q 4

(2 ) ⎢ Fp,q, (x ) 2 ⎢ ⎢ F (2 ) ( x ) ⎢ p,q,3 ⎢ (2 ) ⎣ Fp,q,4 (x )

⎥ ⎥ ⎦

0⎥

5 p 4

1

1

⎤

⎡ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎣ ⎦

(2 ) −Fp,q, (x ) 5

⎥ ⎥

−1

⎢ 2p ⎢ ⎢ ⎢ 3 p2 + 2q ⎢ ⎢ =⎢ ⎢ 4 p3 + 6qp ⎢ ⎢ ⎢ ⎢−5 p4 − 12qp2 − 3q2 ⎣

1

⎤

0

1 2p 3 p + 2q 2

4 p3 + 6 pq

and

0

0

−1 3 − p 2

0

0

−1

0

0⎥ ⎥

−1

0⎥ ⎥

5 p 4

1⎥

5 q 3 5 3 13 p + qp 2 3 −2 p2 −

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

−5 p4 − 3q2 − 12 p2 q

4 − p 3 5 2 3 p + q 3 2

0

⎤

0

0⎥ ⎥

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

321

322

A. S¸ ahin, J.L. Ramírez / Applied Mathematics and Computation 281 (2016) 314–322

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