1 Introduction - DergiPark

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Received : 25.11.2017 Published : 13.12.2017

ISSN: 2149-1402

Year : 2017, Number : 19, Pages: 1-19 Original Article

A MATRIX REPRESENTATION OF A GENERALIZED FIBONACCI POLYNOMIAL Ashok Dnyandeo Godase1,* Macchindra Baburao Dhakne2 1

2

Department of Mathematics, V. P. College, Vaijapur, Maharashtra, India Department of Mathematics, Dr. B. A. M. University, Aurangabad, Maharashtra, India

Abstaract − The Fibonacci polynomial Fn (x) defined recurrently by Fn+1 (x) = xFn (x)+Fn−1 (x), with F0 (x) = 0, F1 (0) = 1, for n ≥ 1 is the topic of wide interest for many years. In this article, b n+1 (x) are introduced and defined by Fbn+1 (x) = generalized Fibonacci polynomials Fbn+1 (x) and L 2 b n+1 (x) = xL b n (x)+L b n−1 (x) with xFbn (x)+Fbn−1 (x) with Fb0 (x) = 0, Fb1 (x) = x +4, for n ≥ 1 and L 2 3 b b L0 (x) = 2x + 8, L1 (x) = x + 4x, for n ≥ 1. Also some basic properties of these polynomials are obtained by matrix methods. Keywords − Fibonacci Sequence, Fibonacci Polynomial, Generalised Fibonacci Polynomial.

1

Introduction

Horadam [9] introduced and studied the generalized Fibonacci sequence Wn = Wn (a; b; p; q) defined by Wn = pWn−1 − qWn−2 with W0 = a, W1 = b, for n ≥ 1 where a; b; p and q are arbitrary complex numbers with q 6= 0. These numbers were first studied by Horadam and they are called Horadam numbers. In [7] Silvester shows that a number of the properties of the Fibonacci sequence can be derived from a matrix representation. In [27] Demirturk obtained summation formulae for the Fibonacci and Lucas sequences by matrix methods. In [28] the authors presented some important relationship between k-Jacobsthal matrix sequence and k-Jacobsthal-Lucas matrix sequence. In [22] Godase and Dhakne described some properties of k- Fibonacci and k-Lucas numbers by matrix terminology. In [18] Catarino and Vasco introduced a 2 × 2 matrix for the k-Pell sequence with its nth power. The well-known Fibonacci polynomial is studied over several years. Many authors are dedicated to study this polynomial. The most research on Fibonacci polynomials are dedicated to study the generalizations of Fibonacci *

Corresponding Author.

Journal of New Theory 19 (2017) 1-19

2

polynomials [4],[5], [20], [19], [21], [28]. The main aim of the present paper is to study other generalized Fibonacci polynomial by matrix methods. Somnuk Srisawat and Wanna Sriprad [1] investigated generalization of Pell and Pell-Lucas numbers, which is called (s, t)-Pell and (s, t)-Pell-Lucas numbers, also they defined the 2 × 2 matrix · ¸ s 2(s2 + t) W = 1 s 2 using this matrix they established many identities of (s, t)-Pell and (s, t)-Pell-Lucas numbers. Hasan Huseyin Gulec, Necati Taskara [2] established a new generalizations for (s, t)-Pell (s, t)- Pell Lucas {qn (s, t)}n ∈ N sequences for Pell and Pell Lucas numbers. Considering these sequences, they defined the matrix sequences which have elements of {pn (s, t)}n ∈ N and {qn (s, t)}n ∈ N . Yuan, Yi, and Wenpeeg Zhang [3] introduced different methods to calculate the summations involving the Fibonacci polynomials. Fikri Koken and Durmus Bozkurt [6] defined the Jacobsthal Lucas E-matrix and R-matrix alike to the Fibonacci Qmatrix. Using this matrix representation they found some equalities and Binet-like formula for the Jacobsthal and Jacobsthal-Lucas numbers. Falcon and Plaza[10] presented the derivatives of Fibonacci polynomials in the form of convolution of k-Fibonacci polynomials and many relations for the derivatives of these polynomials are proved. bn (x) respectively. We denote the Fibonacci and Lucas polynomial by Fbn (x) and L Most of the identities for Fibonacci and Lucas polynomials can be found in the articles [11], [12], [13], [14], [15], [16], [17], [23], [24], [25], [26] on Fibonacci and Lucas sequences and their applications. The ultimate aim of this paper is to introduce new bn (x) of Fibonacci and Lucas polynomials and establish a generalization Fbn (x) and L bn (x) using matrix method. collection of identities for the Fbn (x) and L

2

bn(x) Generalized Fibonacci polynomials Fbn(x) and L

In [8] the Fibonacci polynomial Fn (x) defined recurrently by Fn+1 (x) = xFn (x) + Fn−1 (x), with F0 (x) = 0, F1 (0) = 1, for n ≥ 1. In this paper we defined generalized bn (x). Fibonacci polynomials Fbn (x) and L Definition 2.1. The generalized Fibonacci polynomial Fbn (x) is defined by the recurrence relation Fbn+1 (x) = xFbn (x) + Fbn−1 (x) with Fb0 (x) = 0, Fb1 (x) = x2 + 4, for n ≥ 1

(1)

bn (x) is defined by the recurrence Definition 2.2. The generalized Lucas polynomial L relation bn+1 (x) = xL bn (x) + L bn−1 (x) with L b0 (x) = 2x2 + 8, L b1 (x) = x3 + 4x, for n ≥ 1 L (2)

3

Journal of New Theory 19 (2017) 1-19 Polynomial Fibonacci Lucas Pell Pell-Lucas Jacobsthal Jacobsthal-Lucas Generalized Fibonacci Generalized Lucas

Initial value G0 (x) = a(x) 0 2 0 2 0 2 0 2x2 + 8

Initial value G1 (x) = b(x) 1 x 1 2x 1 1 x2 + 4 x3 + 4x

Recursive Formula Gn+1 (x) = a(x)Gn (x) + b(x)Gn−1 (x) Fn+1 (x) = Fn (x) + Fn−1 (x) Ln+1 (x) = Ln (x) + Ln−1 (x) Pn+1 (x) = 2xPn (x) + Pn−1 (x) Qn+1 (x) = 2xQn (x) + Qn−1 (x) Jn+1 (x) = Jn (x) + 2xJn−1 (x) jn+1 (x) = jn (x) + 2xjn−1 (x) Fbn+1 (x) = xFbn (x) + Fbn−1 (x) b n+1 (x) = xL b n (x) + L b n−1 (x) L

Table 1: Recurrence relation of some GFP. Characteristic equation of the initial recurrence relation (1 and 2) is, r2 − xr − 1 = 0

(3)

Characteristic roots of (3) are r1 (x) =

x+

√

√ x2 + 4 x − x2 + 4 , r2 (x) = 2 2

Characteristic roots (4) satisfy the properties p √ r1 (x) − r2 (x) = x2 + 4 = ∆(x), r1 (x) + r2 (x) = x, r1 (x)r2 (x) = −1

(4)

(5)

bn (x) are given by Binet forms for both Fbn (x) and L

bk,n L

Fbn (x) = r1 (x)n − r2 (x)n £ ¤ = r1 (x)2 + r2 (x)2 + 2 [r1 (x)n + r2 (x)n )]

The most commonly used matrix in relation to the recurrence relation(3) is ¸ · x 1 M (x) = 1 0 Using Principle of Mathematical induction the matrix M can be generalized to  b Fn+1 (x)  ∆(x) M (x)n =   Fbn (x) ∆(x)

Fbn (x) ∆(x) b Fn−1 (x) ∆(x)

   where, n is an integer 

(6) (7)

(8)

Journal of New Theory 19 (2017) 1-19

3

4

Auxiliary Results

bn (x) can be established using (6), (7). Some of Several identities for Fbn (x) and L these are listed below bn (x) = Fbn+1 (x) + Fbn−1 (x) L bn−1 (x) ∆(x)Fbn (x) = Fbn+1 (x) + L ∆(x)Fbn2 (x) = Fb2n (x) − 2∆(x)(−1)n bn (x) = ∆Fbm+n (x) − ∆(x)(−1)m L bn−m (x) Fbm (x)L bm+n (x) − (−1)m L bn−m (x) Fbm (x)Fbn (x) = L (−1)n−m+1 Fbm (x)2 = Fbm+n (x)Fbn−m (x) − Fbn (x)2

(10) (11) (12) (13) (14)

bm+n (x)L bn−m (x) − L bn (x)2 (−1)n−m Fbm (x)2 = L Fbm (x)Fbn+r+m (x) = Fbm+n (x)Fbr+m (x) − (−1)m Fbn (x)Fbr (x)

(15)

b(m+1)n (x) = L bmn (x)L bn (x) + ∆(x)Fbmn (x)Fbn (x) 2∆(x)L bn (x) + L bmn (x)Fbn (x) 2∆(x)Fb(m+1)n (x) = Fbmn (x)L

(17)

bm+n (x)2 + (−1)m−1 L bn (x)2 ∆(x)Fb2n+m (x)Fbm (x) = L bm+n (x)L bn (x) + (−1)m+1 L bn−m (x)L bn (x) ∆(x)Fb2n (x)Fbm (x) = L Fb2(r+1)n+m (x)Fbm (x) = Fbm+2rn (x)Fb2n+m (x) + (−1)m+1 Fb2rn (x)Fb2n (x) Fb−n (x) = (−1)n+1 Fbn (x) Fb−n−1 (x) = (−1)n Fbn+1 (x)

4

(9)

(16) (18) (19) (20) (21) (22) (23)

Main Results

Lemma 4.1. If X is a square matrix with ∆(x)X 2 = xX + I, then ∆(x)X n = Fbn (x)X + Fbn−1 (x)I, for all n ∈ Z Proof. For n = 0 the result is true For n = 1 ∆(x)(X)1 = Fb1 (x)X + Fb0 (x)I = ∆(x)X + 0I = ∆(x)X Hence result is true for n = 1. Assume that, ∆(x)X n = Fbn (x)X+Fbn−1 (x)I, and prove that, ∆(x)X n+1 = Fbn+1 (x)X+ Fbn (x)I

5

Journal of New Theory 19 (2017) 1-19

Consider, Fbn+1 (x)X + Fbn (x)I = (Fbn (x)x + Fbn−1 (x)I)X + Fbn (x)I = (xX + I)Fbn (x) + X Fbn−1 (x) = X 2 Fbn (x) + X Fbn−1 (x) = X(X Fbn (x) + Fbn−1 (x))

Hence, ∆(x)X n+1

= X(∆(x)X n ) = ∆(x)X n+1 = Fbn+1 (x)X + Fbn (x)I

We now show that, ∆(x)X −n = Fb−n (x)X + Fb−n−1 (x)I, for all n ∈ Z + Let, Y = xI − X, then Y 2 = (xI − X)2 = x2 I − 2xX + X 2 = x2 I − 2xX + xX + I = x2 I − xX + I = x(xI − X) + X + I = xY + I This shows that ∆Y n = Fbn (x)Y + Fbn−1 (x)I ∆(x)(−X −1 )n = Fbn (x)(xI − X) + Fbn−1 (x)I ∆(x)(−1)n X −n = −Fbn (x)X + Fbn+1 (x)I ∆(x)X −n = (−1)n+1 Fbn (x)X + (−1)n Fbn+1 (x)I Using equations (22 and 23), it gives that ∆(x)X −n = Fb−n (x)X + Fb−n−1 (x)I, for all n ∈ Z+ # " b · x ∆(x) ¸ ∆Fbn (x) Ln (x) 2 Corollary 4.2. Let, S(x) = 21 , then ∆(x)S(x)n = Fbn2(x) Lbn2(x) , for x 2

2

2

every n ∈ Z Proof.

· 2

Since S(x) =

x2 +2 2 x 2

x∆(x) 2 x2 +2 2

2

¸

= xS(x) + I Using Lemma (4.1), it is clear that ∆(x)S(x)n = Fbn (x)S(x) + Fbn−1 (x)I " b # " b = " =

xFn (x) 2 Fbn (x) 2 b n (x) L 2 Fbn (x) 2

∆(x)Fn (x) 2 xFbn (x) 2 # ∆(x)Fbn (x) 2 b n (x) L 2

+

Fbn−1 (x) 0 b 0 Fn−1 (x)

#

6

Journal of New Theory 19 (2017) 1-19

Lemma 4.3. b2 (x) − ∆(x)Fb2 (x) = 4∆(x)2 (−1)n L n n

for all, n ∈ Z

(24)

Proof. Since,

det(S(x)) = −1 det(S(x)n ) = (−1)n " b

Moreover since, ∆(x)S(x)n =

Ln (x) 2 Fbn (x) 2 2 b Ln (x)

∆Fbn (x) 2 b n (x) L 2

#

∆(x)Fbn2 (x) 4 4 2 n 2 2 b b Thus it follows that, Ln (x) − ∆(x)Fn (x) = 4∆(x) (−1) for all, n ∈ Z We get,

det(∆(x)S(x)n ) =

−

Lemma 4.4. bn+m (x) = L bn (x)L bm (x) + ∆(x)Fbn (x)Fbm (x) for all n, m ∈ Z 2∆(x)L

(25)

bm (x) + L bn (x)Fbm (x) for all n, m ∈ Z 2∆(x)Fbn+m (x) = Fbn (x)L

(26)

Proof. Since, ∆(x)2 S(x)n+m = ∆(x)S(x)n .∆(x)S(x)m " b #" b b = " = " But, ∆(x)S(x)n+m =

#

Ln (x) Lm (x) ∆(x)Fn (x) ∆(x)Fbm (x) 2 2 2 2 . Fbm (x) b n (x) b m (x) Fbn (x) L L 2 2 2 2 b n (x)L b m (x)+∆(x)Fbn (x)Fbm (x) b n (x)Fbm (x)+Fbn (x)L b m (x) L ∆(x)[L 4 4 b n (x)Fbm (x)+Fbn (x)L b m (x) b n (x)L b m (x)+∆(x)Fbn (x)Fbm (x) L L 4 4 # b n+m (x) L ∆(x)Fbn+m (x) 2 2 b n+m (x) Fbn+m (x) L 2 2

bn+m (x) = L bn (x)L bm (x) + ∆(x)Fbn (x)Fbm (x) for all n, m ∈ Z It gives that 2∆(x)L bm (x) + L bn (x)Fbm (x) for all n, m ∈ Z 2∆(x)Fbn+m (x) = Fbn (x)L

Lemma 4.5. bn−m (x) = L bn (x)L bm (x) − ∆(x)Fbn (x)Fbm (x) for all n, m ∈ Z (27) 2(−1)m ∆(x)2 L bm (x) − L bn (x)Fbm (x) for all n, m ∈ Z 2(−1)m ∆(x)2 Fbn−m (x) = Fbn (x)L

(28)

#

7

Journal of New Theory 19 (2017) 1-19

Proof. Since ∆(x)2 S(x)n−m = ∆(x)S(x)n .∆(x)S(x)−m = ∆(x)S(x)n .∆(x)[S(x)m ]−1 " b = = =

#

Lm (x) −∆(x)Fbm (x) n m 2 2 ∆(x)S(x) .(−1) b m (x) −Fbm (x) L 2 " b #" b 2 # Ln (x) ∆(x)Fbn (x) Lm (x) −∆(x)Fbm (x) 2 2 (−1)m Fbn2(x) . −Fbm2 (x) b n (x) b m (x) L L 2 2 " b b 2 b b2 # b n (x)Fbm (x)−Fbn (x)L b m (x) Ln (x)Lm (x)−∆Fn (x)Fm (x) ∆(x)[L 4 4 b n (x)L b m (x)−∆(x)Fbn (x)Fbm (x) b n (x)Fbm (x)−Fbn (x)L b m (x) L L 4 4

But

" 2

n−m

∆(x) S(x)

=

Ln−m (x) 2 Fn−m (x) 2

∆(x)Fn−m (x) 2 Ln−m (x) 2

#

It gives bn−m (x) = L bn (x)L bm (x) − ∆(x)Fbn (x)Fbm (x) for all n, m ∈ Z 2(−1)m ∆(x)2 L bm (x) − L bn (x)Fbm (x) for all n, m ∈ Z 2(−1)m ∆(x)2 Fbn−m (x) = Fbn (x)L Lemma 4.6. bn−m (x) + ∆(x)L bn+m (x) = L bn (x)L bm (x) (−1)m ∆(x)L

(29)

bm (x) (−1)m ∆(x)Fbn−m (x) + ∆(x)Fbn+m (x) = Fbn (x)L

(30)

Proof. " =

∆(x)2 S(x)n+m + (−1)m ∆(x)2 S(x)n−m # mb b b

b n+m (x)+(−1)m ∆(x)Ln−m (x) ∆(x)L 2 ∆(x)Fbn+m (x)+(−1)m ∆(x)Fbn−m (x) 2

∆(x)[Fn+m (x)+(−1) Fn−m (x) 2 b n+m (x)+(−1)m ∆(x)L b n−m (x) ∆(x)L 2

On the other hand ∆(x)2 S(x)n+m + (−1)m ∆(x)2 S(x)n−m = ∆(x)S(x)n ∆(x)S(x)m + (−1)m ∆(x)S(x)n ∆(x)S(x)−m = ∆(x)S(x)n [∆(x)S(x)m + (−1)m ∆(x)S(x)−m ] " = " = " =

# ("

b m (x) bn ∆(x)Fbn (x) L L 2 2 2 b n (x) L Fbn (x) Fbm (x) 2 2 2 #" b b Ln (x) ∆(x)Fn (x) bm (x) L 2 2 . b b Fn (x) Ln (x) 0 2 2 # b m (x)L b n (x) b m (x) L ∆(x)Fbn (x)L 2 2 b m (x) b m (x)L b n (x) Fbn (x)L L 2 2

∆(x)Fbm (x) 2 b m (x) L 2

#

0 b Lm (x)

"

# +

b m (x) L 2 −Fbm 2

−∆(x)Fbm (x) 2 b m (x) L 2

#)

8

Journal of New Theory 19 (2017) 1-19

It gives bn−m (x) + ∆(x)L bn+m (x) = L bn (x)L bm (x) (−1)m ∆(x)L bm (x) (−1)m ∆(x)Fbn−m (x) + ∆(x)Fbn+m (x) = Fbn (x)L

Lemma 4.7. bt (x)L by (x)Fbz (x) + Fbt (x)L by (x)L bz (x) 8∆(x)2 Fbt+y+z (x) = L bt (x)Fby (x)L bz (x) + ∆(x)Fbt (x)Fby (x)Fbz (x) +L

(31)

bt+y+z (x) = L bt (x)L by (x)L bz (x) 8∆(x)2 L h i by (x)Fbz (x) + Fbt (x)Fby (x)L bz (x) bt (x)Fby (x)Fbz (x) + Fbt (x)L +∆(x) L

(33)

(32)

(34)

Proof. Since " 2

t+y+z

∆(x) S(x)

=

∆(x)Lt+y+z (x) 2 ∆(x)Ft+y+z (x) 2

∆(x)2 Ft+y+z (x) 2 ∆(x)Lt+y+z (x) 2

#

On the other hand ∆(x)2 S(x)t+y+z = ∆(x)S(x)t+y ∆(x)S(x)z #" " b b = " =

#

b z (x) ∆(x)Ft+y (x) Lt+y (x) L ∆(x)Fbz (x) 2 2 2 2 . Fbz (x) b t+y (x) b z (x) Fbt+y (x) L L 2 2 2 2 b t+y (x)L b z (x)+∆(x)Fbt+y (x)Fbz (x) b t+y (x)Fbz (x)+Fbt+y (x)L b z (x) L ∆(x)[L 4 b z (x)Fbt+y (x)+Fbz (x)L b t+y (x) L 4

#

4 b t+y (x)L b z (x)+∆(x)Fbt+y (x)Fbz (x) L 4

Using equations (25) and (26), it gives that bt (x)L by (x)Fbz (x) + Fbt (x)L by (x)L bz (x) 8∆(x)2 Fbt+y+z (x) = L bt (x)Fby (x)L bz (x) + ∆(x)Fbt (x)Fby (x)Fbz (x) +L bt+y+z (x) = L bt (x)L by (x)L bz (x) 8∆(x)2 L h i b b b b b b b b b +∆(x) Lt (x)Fy (x)Fz (x) + Ft (x)Ly (x)Fz (x) + Ft (x)Fy (x)Lz (x)

Theorem 4.8. 2 bt+y (x)Fby+z (x) − ∆(x)(−1)t+z (x)Fby+z b2t+y (x) − (−1)t+y+1 Fbz−t (x)L (x) L b2z−t (x) = (−1)y+z L

(35)

9

Journal of New Theory 19 (2017) 1-19

b2 (x) − (−1)x+z L bz−t (x)L bt+y (x)L by+z (x) + (−1)x+z ∆(x)L b2 (x) ∆(x)L t+y y+z y+z+1 2 b2 = (−1) ∆(x) F (x)

(36)

2 2 bt−z (x)Fbt+y (x)Fby+z (x) + ∆(x)(−1)x+z Fby+z ∆(x)Fbt+y (x) − L (x) 2 = (−1)y+z ∆(x)Fbz−t ,

(37)

z−t

for all t, y, z ∈ Z, t 6= z Proof. Consider the matrix multiplication " b #" # " # Lt (x) ∆(x)Fbt (x) by (x) bt+y (x) L ∆(x) L 2 2 = b z (x) Fbz (x) L Fby (x) ∆(x)Fby+z (x) 2 2 # " b ∆(x)Fbt (x) Lt (x) bt (x)L bz (x) − ∆(x)Fbt Fbz (x) L 2 = det Fbz2(x) b Lz (x) 4 2

2

bz−t (x) (−1)t ∆(x)L 2 =R 6= 0 =

Hence

"

by (x) L Fby (x)

#

" = =

by (x) = L

#−1 " # b t (x) L ∆(x)Fbt (x) b ∆(x)Lt+y (x) 2 2 . b z (x) Fbz (x) L ∆(x)Fby+z (x) 2 "2b #" Lz (x) −∆(x)Fbt (x) bt+y (x) 1 ∆(x)L 2 2 b b Lt (x) R −Fz (x) ∆(x)Fby+z (x) 2 2 bz (x)L bt+y (x) − ∆(x)Fbt Fby+z (x)] (−1)t [L

#

bz−t (x) L bt (x)Fbz+y (x) − Fbz (x)L by+t (x)] (−1)t [L Fby (x) = bz−t (x) L

Since b2 (x) − ∆(x)Fb2 (x) = 4∆(x)2 (−1)y L y y We get bz (x)L bt+y (x) − ∆(x)Fbt (x)Fby+z (x)]2 − ∆(x)[L bt (x)Fbz+y (x) − Fbz (x)L by+t (x)]2 [L b2z−t = 4(−1)y ∆(x)2 L

10

Journal of New Theory 19 (2017) 1-19

Using equations (27), (28), (31) and (33) b2 (x)L b2 (x) − 2∆(x)L bz (x)Fbt+y (x)Fby+z (x) + ∆(x)2 Fb2 (x)Fb2 (x))− (L z t+y t y+z 2 2 2 b (x)Fb (x) − 2L bt (x)Fbz (x)Fby+z (x)L bt+y (x) + Fb (x)L b2 (x)) ∆(x)(L t

y+z

z

y

2

= 4(−1) ∆(x)

t+y L2z−t (x)

It gives that b2 (x) − (−1)t+y+1 Fbz−t (x)L bt+y (x)Fby+z (x) − ∆(x)(−1)t+z Fb2 (x) = (−1)y+z L b2 (x) L t+y y+z z−t for all t, y, z ∈ Z

Proof of (36) and (37) is similar to (35). Theorem 4.9. For n ∈ N and m, t ∈ Z with m 6= 0 j=n X

m b b b b bmj+t (x) = ∆(x)Lt (x) − ∆(x)Lmn+m+t (x) + (−1) ∆(x)(Lmn+t (x) − Lt−m (x)) L bm (x) ∆(x) + (−1)m ∆(x) − L j=0 (38) j=n X ∆(x)Fbt (x) − ∆(x)Fbmn+m+t (x) + (−1)m ∆(x)(Fbmn+t (x) − Fbt−m (x)) Fbmj+t (x) = bm (x) ∆(x) + (−1)m ∆(x) − L j=0

(39) Proof. Since

2

2

m n+1

∆(x) I − ∆(x) (S(x) )

m

= (∆(x)I − ∆(x)S(x) )

j=n X

∆(x)(S(x)m )j

j=0

From Lemma (4.3) it is clear that det(∆(x)I − ∆(x)S(x)m ) 6= 0, therefore we get (∆(x)I − ∆(x)(S(x)m ))−1 (∆(x)I − ∆(x)(S(x)m )n+1 )∆(x)S(x)t j=n X = (∆(x)S(x)mj+t )

" P j=n b 1 (Lmj+t (x)) = 21 Pj=0 j=n b j=0 (Fmj+t (x)) 2

j=0

∆(x) Pj=n b (Fmj+t (x)) 2 P j=0 j=n b 1 j=0 (Lmj+t (x)) 2

#

bm (x), then we can write For m 6= 0, take D = ∆(x) + (−1)m ∆(x) − L " # b m (x)) (L (Fbm (x)) 1 ∆(x) − ∆(x) 2 2 (∆(x)I − ∆(x)S(x)m )−1 = b m (x)) (Fbm (x)) (L D ∆(x) − 2 2 # " bm (x)) (L (Fbm (x)) 1 (∆(x) − )I + R = D 2 2

11

Journal of New Theory 19 (2017) 1-19

It gives that, (∆(x)I − ∆(x)S(x)m )−1 (∆(x)2 S(x)t − ∆(x)2 S(x)mn+m+t ) " # bm (x)) 1 (L (Fbm (x)) = (∆(x) − )I + R (∆(x)2 S(x)t − ∆(x)2 S(x)mn+m+t ) D 2 2 (∆(x)I − ∆(x)S(x)m )−1 (∆(x)2 S(x)t − ∆(x)2 S(x)mn+m+t ) " # bm (x)) 1 (L = (∆(x) − )(∆(x)2 S(x)t − ∆(x)2 S(x)mn+m+t ) D 2 " # 1 (Fbm (x)) + R(∆(x)2 S(x)t − ∆(x)2 S(x)mn+m+t ) D 2 From Corollary (4.2) and (4.3), we get R(∆(x)2 S(x)t − ∆(x)2 S(x)mn+m+t ) = " # b b b t (x)−L b mn+m+t (x)) (x)) (L ∆(x) (Ft (x)−Fmn+m+t ∆(x) 2 2 ∆(x) b t (x)−L b mn+m+t (x)) (L (Fbt (x)−Fbmn+m+t (x)) ∆(x) 2 2 Hence, we get bm (x) L ∆(x) (∆(x)I − ∆(x)S(x)m )−1 (∆(x)2 S(x)t − ∆(x)2 S(x)mn+m+t ) = (∆(x) − ) D # 2 " b b b b (x)) (x)) ∆(x)Fbm (x) ∆(x) (Lt (x)−Lmn+m+t ∆(x) (Ft (x)−Fmn+m+t 2 2 + b t (x)−L b mn+m+t (x)) (L (Fbt (x)−Fbmn+m+t (x)) 2D 2 2 " # b t (x)−L b mn+m+t (x)) ∆(x)(Fbt (x)−Fbmn+m+t (x)) (L ∆(x) 2 2 b t (x)−L b mn+m+t (x)) (L (Fbt (x)−Fbmn+m+t (x)) ∆(x) 2 2 Hence, it gives that j=n X

"

bmj+t (x) L

j=0

# bm (x) bm (x) ∆(x)2 F L bt (x) − L bmn+m+t (x)) + = )(L (Fbt (x) − Fbmn+m+t (x) (∆(x) − D 2 2 j=n X

"

j=0

Fbmj+t (x)

# bm (x) bm (x) F ∆(x)2 L bt (x) − L bmn+m+t (x) )(Fbt (x) − Fbmn+m+t (x)) + (L = (∆(x) − D 2 2 Using (31) and (33), we get

12

Journal of New Theory 19 (2017) 1-19

j=n X

bt (x) − ∆(x)L bmn+m+t (x) + (−1)m ∆(x)(L bmn+t (x) − L bt−m (x)) ∆(x)L b Lmj+t (x) = bm (x) ∆(x) + (−1)m ∆(x) − L j=0

j=n X

∆(x)Fbt (x) − ∆(x)Fbmn+m+t (x) + (−1)m ∆(x)(Fbmn+t (x) − Fbt−m (x)) Fbmj+t (x) = bm (x) ∆(x) + (−1)m ∆(x) − L j=0

Theorem 4.10. For n ∈ N and m, t ∈ Z j=n m X b b b b bmj+t (x) = ∆(x)Lt (x) − ∆(x)Lmn+m+t (x) + (−1) ∆(x)(Lmn+t (x) − Lt−m (x)) (−1)j L bm (x) ∆(x) + (−1)m ∆(x) − L j=0 (40) j=n X ∆(x)Fbt (x) − ∆(x)Fbmn+m+t (x) + (−1)m ∆(x)(Fbmn+t (x) − Fbt−m (x)) (−1)j Fbmj+t (x) = bm (x) ∆(x) + (−1)m ∆(x) − L j=0

(41) Proof. Case: 1 If n is an even natural number then we have 2

2

m n+1

∆(x) I + ∆(x) (S(x) )

j=n X = (∆(x)I + ∆(x)S(x) ) (∆(x)S(x)m )j (−1)j m

j=0

From Lemma (4.3) it is clear that det(∆(x)I + ∆(x)S(x)m ) 6= 0, therefore we get ∆(x)I + ∆(x)(S(x)m )−1 (∆(x)I + ∆(x)(S(x)m )n+1 )∆(x)S(x)t j=n X = (−1)j ∆(x)(S(x)mj+t )

" P j=n 1 bmj+t (x)) (−1)j (L 2 = 1 Pj=0 j=n j b j=0 (−1) (Fmj+t (x)) 2

j=0

∆(x) Pj=n (−1)j (Fbmj+t (x)) 2 P j=0 j=n 1 j b j=0 (−1) (Lmj+t (x)) 2

#

bm (x), then we get For m 6= 0 take d = ∆(x) + ∆(x)(−1)m + L " # b m (x)) (Fbm (x)) (L 1 −∆(x) ∆(x) + 2 2 (∆(x)I + ∆(x)S(x)m )−1 = b m (x)) (−Fbm (x)) (L d ∆(x) + 2 2 " # bm (x)) 1 (L (Fbm (x)) = (∆(x) + )I − R d 2 2 Thus, it is seen that (∆(x)I + ∆(x)S(x)m )−1 (∆(x)2 S(x)t + ∆(x)2 S(x)mn+m+t ) " # bm (x)) (Fbm ) 1 (L )I − R = (∆(x) + d 2 2 (∆(x)2 S(x)t + ∆(x)2 S(x)mn+m+t )

13

Journal of New Theory 19 (2017) 1-19

By Corollary (4.2) and (4.3), we get R(∆(x)2 S(x)t + ∆(x)2 S(x)mn+m+t ) " # b mn+m+t (x)) b b b t (x)+L (x)) (L ∆(x) ∆(x) (Ft (x)+Fmn+m+t 2 2 = ∆(x) b t (x)+L b mn+m+t (x)) (L (Fbt (x)+Fbmn+m+t (x)) ∆(x) 2 2 Thus, it follows that bm (x) ∆(x) L (∆(x)I + ∆(x)S(x)m )−1 (∆(x)2 S(x)t + ∆(x)2 S(x)mn+m+t ) = (∆(x) + ) d 2 # " b b b b (x)) (x)) ∆(x) (Lt (x)+Lmn+m+t ∆(x) (Ft (x)+Fmn+m+t 2 2 ∆(x)Fbm (x) − 2d

"

(Fbt (x)+Fbmn+m+t (x)) 2 ∆(x)(Fbt (x)+Fbmn+m+t (x)) ∆(x) 2 b t (x)+L b mn+m+t (x)) (L 2

b t (x)+L b mn+m+t (x)) (L 2 b t (x)+L b mn+m+t (x)) (L ∆(x) 2 (Fbt (x)+Fbmn+m+t (x)) ∆(x) 2

#

Thus, it gives that j=n X

bmj+t (x) (−1)j L

j=0

"

# bm (x) bm (x)∆(x) L F ∆(x)2 bt (x) + L bmn+m+t (x)) − = (∆(x) + )(L (Fbt (x) + Fbmn+m+t (x) d 2 2 j=n X (−1)j Fbmj+t (x) j=0

"

# bm (x) bm (x)∆(x) L F ∆(x)2 bt (x) + L bmn+m+t (x) = (∆(x) + )(Fbt (x) + Fbmn+m+t (x)) − (L d 2 2 Using (31) and (33), we get j=n X bmj+t (x) (−1)j L j=0

bt (x) − ∆(x)L bmn+m+t (x) + (−1)m ∆(x)(L bmn+t (x) − L bt−m (x)) ∆(x)L = bm (x) ∆(x) + (−1)m ∆(x) − L j=n X

(−1)j Fbmj+t (x)

j=0

=

∆(x)Fbt (x) − ∆(x)Fbmn+m+t (x) + (−1)m ∆(x)(Fbmn+t (x) − Fbt−m (x)) bm (x) ∆(x) + (−1)m ∆(x) − L

Case: 2 If n is an odd natural number, we get j=n j=n−1 X X jb bmj+t (x) − L bmn+t (x) (−1) Lmj+t (x) = (−1)j L j=0

j=0

14

Journal of New Theory 19 (2017) 1-19

Since n is an odd natural number then (n − 1) is an even number, hence using case-I, it follows that, j=n X

b mj+t (x) = (−1)j L

j=0

=

b t (x) + ∆(x)L b mn+t (x) + (−1)m ∆(x)(L b mn−m+t (x) + L b t−m (x)) ∆(x)L b mn+t (x) −L m b ∆(x) + (−1) ∆(x) + Lm (x)

b t (x) + (−1)m ∆(x)(L b mn−m+t (x) + L b t−m (x)) − (−1)m ∆(x)L b mn+t (x) − ∆(x)L b m (x)L b mn+t (x) ∆(x)L m b ∆(x) + (−1) ∆(x) + Lm (x)

Using equations (31) and (33), we get j=n m X b b b b bmj+t (x) = ∆(x)Lt (x) + ∆(x)Lmn−m+t (x) + (−1) ∆(x)(Lt−m (x) − Lmn+t (x)) (−1)j L bm (x) ∆(x) + (−1)m ∆(x) + L j=0 j=n X bmn+m+t (x) + (−1)m ∆(x)(Fbt−m (x) − Fbmn+t (x)) ∆(x)Fbt (x) − ∆(x)L (−1)j Fbmj+t (x) = bm (x) ∆(x) + (−1)m ∆(x) + L j=0

5

Tables n

bn (x) F

b n (x) L

1

4 + x2

4x + x3

2

4x + x3

8 + 6x2 + x4

3

4 + 5x2 + x4

12x + 7x3 + x5

4

8x + 6x3 + x5

8 + 18x2 + 8x4 + x6

5

4 + 13x2 + 7x4 + x6

20x + 25x3 + 9x5 + x7

6

12x + 19x3 + 8x5 + x7

8 + 38x2 + 33x4 + 10x6 + x8

7

4 + 25x2 + 26x4 + 9x6 + x8

28x + 63x3 + 42x5 + 11x7 + x9

8

16x + 44x3 + 34x5 + 10x7 + x9

8 + 66x2 + 96x4 + 52x6 + 12x8 + x10

9

4 + 41x2 + 70x4 + 43x6 + 11x8 + x10

36x + 129x3 + 138x5 + 63x7 + 13x9 + x11

10

20x + 85x3 + 104x5 + 53x7 + 12x9 + x11

8 + 102x2 + 225x4 + 190x6 + 75x8 + 14x10 + x12

11

4 + 61x2 + 155x4 + 147x6 + 64x8 + 13x12 + x12

44x + 231x3 + 363x5 + 253x7 + 88x9 + 15x11 + x13

12

24x + 146x3 + 259x5 + 200x7 + 76x9 + 14x11 + x13

8 + 146x2 + 456x4 + 553x6 + 328x8 + 102x10 + 16x12 + x14

b n (x) Table 2: First 12 terms of Fbn (x) and L

Fbn (x) x=1 x=2 x=3 x=4 x=5 x=6 x=7 x=8 x=9 x = 10 x = 11 x = 12 x = 13 x = 14 x = 15 x = 16 x = 17 x = 18 x = 19 x = 20

Fb1 (x) 5 8 13 20 29 40 53 68 85 104 125 148 173 200 229 260 293 328 365 404

Fb2 (x) 5 16 39 80 145 240 371 544 765 1040 1375 1776 2249 2800 3435 4160 4981 5904 6935 8080

Fb3 (x) 10 40 130 340 754 1480 2650 4420 6970 10504 15250 21460 29410 39400 51754 66820 84970 106600 132130 162004

Fb4 (x) 15 96 429 1440 3915 9120 18921 35904 63495 106080 169125 259296 384579 554400 779745 1073280 1449471 1924704 2517405 3248160

Fb6 (x) 40 560 4680 25840 105560 346320 964600 2369120 5269320 10819120 20801000 37855440 65760760 109768400 176998680 276902080 421791080 627447600 913811080 1305752240

Fb7 (x) 65 1352 15457 109460 548129 2134120 6887297 19244612 48002305 109262504 230686625 457398292 859918817 1544558600 2666728129 4447672580 7195174337 11328808072 17410373345 26180170004

Fb8 (x) 105 3264 51051 463680 2846205 13151040 49175679 156326016 437290065 1103444160 2558353875 5526634944 11244705381 21733588800, 40177920615, 71439663360 122739754809 204545992896 331710904635 524909152320

Table 3: First few terms of Fbn

Fb5 (x) 25 232 1417 6100 20329 56200 135097 291652 578425 1071304 1875625 3133012 5028937 7801000 11747929 17239300 24725977 34751272 47962825 65125204

Fb9 (x) 170 7880 168610 1964180 14779154 81040360 351117050 1269852740 3983612890 11143704104 28372579250 66777017620 147041088770 305814801800 605335537354 1147482286340 2093771006090 3693156680200 6319917561410 10524363216404

Fb10 (x) 275 19024 556881 8320400 76741975 499393200 2506995029 10315147936 36289806075 112540485200 314656725625 806850846384 1922778859391 4303140814000 9120210980925 18431156244800 35716846858339 66681366236496 120410144571425 211012173480400

Journal of New Theory 19 (2017) 1-19

15

b n (x) L x=1 x=2 x=3 x=4 x=5 x=6 x=7 x=8 x=9 x = 10 x = 11 x = 12 x = 13 x = 14 x = 15 x = 16 x = 17 x = 18 x = 19 x = 20

b 1 (x) L 5 16 39 80 145 240 371 544 765 1040 1375 1776 2249 2800 3435 4160 4981 5904 6935 8080

b 2 (x) L 15 48 143 360 783 1520 2703 4488 7055 10608 15375 21608 29583 39600 51983 67080 85263 106928 132495 162408

b 3 (x) L 20 112 468 1520 4060 9360 19292 36448 64260 107120 170500 261072 386828 557200 783180 1077440 1454452 1930608 2524340 3256240

b 4 (x) L 35 272 1547 6440 21083 57680 137747 296072 585395 1081808 1890875 3154472 5058347 7840400 11799683 17306120 24810947 34857872 48094955 65287208

b 6 (x) L 90 1584 16874 115560 568458 2190320 7022394 19536264 48580730 110333808 232562250 460531304 864947754 1552359600 2678476058 4464911880 7219900314 11363559344 17458336170 26245295208

b 7 (x) L 145 3824 55731 489520 2951765 13497360 50140279 158695136 442559385 1114263280 2579154875 5564490384 11310466141 21843357200 40354919295 71716565440 123161545889 205173440496 332624715715 526214904560

b 8 (x) L 235 9232 184067 2073640 15327283 83174480 358004347 1289097352 4031615195 11252966608 28603265875 67234415912 147901007587 307359360400 608002265483 1151929958920 2100966180427 3704485488272 6337327934755 10550543386408

b n (x) Table 4: First few terms of L

Fb5 (x) 55 656 5109 27280 109475 355440 983521 2405024 5332815 10925200 20970125 38114736 66145339 110322800 177778425 277975360 423240551 629372304 916328485 1309000400

b 9 (x) L 380 22288 607932 8784080 79588180 512544240 2556170708 10471473952 36727096140 113643929360 317215079500 812377481328 1934023564772 4324874402800 9160388901540 18502595908160 35839586613148 66885912229392 120741855476060 211537082632720

b 10 (x) L 615 53808 2007863 3720996 413268183 3158439920 18251199303 85060888968 334575480455 1147692260208 3517969140375 9815764191848 25290207349623 60855600999600 138013835788583 297193464489480 611373938603943 1207650905617328 2300432581979895 4241292196040808

Journal of New Theory 19 (2017) 1-19

16

Journal of New Theory 19 (2017) 1-19

17

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