Laplace - Fibonacci transform by the solution of ...

Nonauton. Dyn. Syst. 2017; 4:22–30

Research Article

Open Access

Sandra Pinelas*, G.B.A Xavier, S.U. Vasantha Kumar, and M. Meganathan

Laplace - Fibonacci transform by the solution of second order generalized difference equation https://doi.org/10.1515/msds-2017-0003 Received June 29, 2016; accepted April 24, 2017

Abstract: The main objective of this paper is finding new types of discrete transforms with tuning factor t. This is not only analogy to the continuous Laplace transform but gives discrete Laplace-Fibonacci transform (LFt ). This type of Laplace-Fibonacci transform is not available in the continuous case. The LFt generates uncountably many outcomes when the parameter t varies on (0, ∞). This possibility is not available in the existing Laplace transform. All the formulae and results derived are verified by MATLAB. Keywords: Generalized difference operator, Two dimensional Fibonacci sequence, Closed form solution, Fibonacci summation formula, Laplace-Fibonacci Transform MSC: 39A70, 39A10, 44A10, 47B39, 65J10, 65Q10

1 Introduction Fibonacci and Lucas numbers cover a wide range of interest in modern mathematics as they appear in the comprehensive works of Koshy [7] and Vajda [18]. The k−Fibonacci sequence introduced by Falcon and Plaza [3] depends only on one integer parameter k and is defined as F k,0 = 0, F k,1 = 1 and F k,n+1 = kF k,n + F k,n−1 , where n ≥ 1, k ≥ 1. In particular, if k = 2, the Pell sequence is obtained as P0 = 0, P1 = 1 and P n+1 = 2P n +P n−1 for n ≥ 1. In 2015, M. Lawrence Glasser and Yajun Zhou [9] report on an Integral representation for the Fibonacci Numbers and their Generalization. In 2015, Martin Griffiths and William Wynn-Thomas [11] construct, by way of the Fibonacci numbers, a geometrical object resembling an infinite staircase and demonstrate an interesting property of this Fibonacci staircase. Koshy [8] construct graph-theoretic models for an extended univariate Fibonacci family, which includes Fibonacci, Lucas, Pell, and Pell-Lucas polynomials. In 2016, Jeremy F. Alm and Taylor Herald [5] define a variant of Fibonacci-like sequences(Prime Fibonacci sequences), where one takes the sum of the previous two terms and returns the smallest odd prime divisor of that sum as the next term and proved some results. Mahadi Ddamulira, Florian Luca and Mihaja Rakotomalala [10] find all Fibonacci numbers which are products of two Pell numbers and all Pell numbers which are products of two Fibonacci numbers. R.S.Melham (refer [14], [15]) find closed forms, in terms of rational numbers, for certain finite sums. His most general results are for finite sums where the denominator of the summand is a product of terms from a se-

*Corresponding Author: Sandra Pinelas: Academia Militar, Departamento de CiÃłncias Exactas e Naturais, Av. Conde Castro GuimarÃčes, 2720-113 Amadora, Portugal, E-mail: [email protected] G.B.A Xavier: Department of Mathematics, Sacred Heart College, Tirupattur - 635601, Vellore District, Tamil Nadu, S. India, E-mail: [email protected] S.U. Vasantha Kumar, M. Meganathan: Department of Mathematics, Sacred Heart College, Tirupattur - 635601, Vellore District, Tamil Nadu, S. India Open Access. Âľ 2017 Sandra Pinelas et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Unauthenticated

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Laplace - Fibonacci transform by the solution of second order generalized difference equation |

23

quence that generalizes both the Fibonacci and Lucas numbers. Also he created a link to the Fibonacci/Lucas numbers then facilitates the derivation of closed forms for reciprocal series that involve the Fibonacci/Lucas numbers. The term that defines the denominator of each summand contains squares of Fibonacci related numbers, with subscripts in arithmetic progression. To develop Two Dimensional Fibonacci sequence and Laplace-Fibonacci transform, we need to reveal basic theory of generalized difference operators ∆` and ∆ α(`) . In 1984, Jerzy Popenda [6] introduced a particular type of difference operator on u(k) as ∆ α u(k) = u(k + 1) − αu(k), In 2011, M.Maria Susai Manuel, et.al, [13] extended the operator ∆ α to generalized α−difference operator as ∆ v(k) = v(k + `) − αv(k) for the real α(`)

valued function v(k). These operators induces to introduce the following second order generalized difference operator. Here, for each pair (a1 , a2 ) ∈ R2 , where R is the set of all real numbers and ` > 0, second order generalized difference operator on v(k) is defined as ∆

(a1 ,a2 )`

v(k) = v(k) − a1 v(k − `) − a2 v(k − 2`),

k ∈ (−∞, ∞),

(1)

which generates two dimensional Fibonacci sequence and its sum. By taking a1 = α, a2 = 0 in (1) we can get the above two operators. For (a1 , a2 ) ∈ R2 , a two dimensional Fibonacci sequence is defined as F(a1 ,a2 ) = {F n }∞ n=0 , where F 0 = 1, F 1 = a 1 , F n = a 1 F n−1 + a 2 F n−2 , n ≥ 2.

(2)

The sequence (2) becomes the well known Fibonacci sequence when a1 = a2 = 1. For example, the two dimensional Fibonacci sequences F(8,16) = {1, 8, 80, 768, · · · } and F(0.7,1.5) = {1, 0.7, 1.99, 2.443, 4.6951, · · · } are obtained from (2) by taking a1 = 8, a2 = 16 and a1 = 0.7, a2 = 1.5 respectively. The Pell sequence F(2,1) = {0, 1, 2, 5, 12, 29, 70, . . .} is obtained by taking F0 = 0, F1 = 1 and F n = 2F n−1 + F n−2 for n ≥ 2. Similarly, one can obtain two dimensional Fibonacci sequence corresponding to each pair (a1 , a2 ) ∈ R2 .

2 Sum of Fibonacci Numbers With Function Values The operator defined in (1) generates several formulae directly on Fibonacci numbers. The method to derive the above is simple. First we need to form a generalized difference equation and then by equating the closed form and summation solutions, we get the desired formula. Hence in this section, we derive some formulae on sum of Fibonacci sequence using inverse of second order difference operator. Definition 2.1. For each pair (a1 , a2 ) ∈ R2 and ` ∈ (0, ∞), consider (1). If

∆

(a1 ,a2 )`

v(k) = u(k),

then we write

v(k) =

−1

∆

u(k).

(a1 ,a2 )`

(3)

h a2 i−1 a1 − 2s` in (3), we get Example 2.2. Taking u(k) = a sk and v(k) = a sk 1 − s` a a −1

∆

(a1 ,a2 )`

a sk =

a sk

a1 a2 a2 , where 1 − a s` − a2s` ≠ 0. a1 1 − s` − 2s` a a

(4)

Hereafter, we assume that F n ∈ F(a1 ,a2 ) and 1 − a1 a2 ≠ 0.

Unauthenticated Download Date | 9/2/17 2:39 PM

24 | Sandra Pinelas et al. Theorem 2.3. (Fibonacci-Function value Summation Formula) Let F n ∈ F(a1 ,a2 ) and v(k) be a solution to the second order generalized difference equation ∆ v(k) = u(k), k ∈ (−∞, ∞). Then we have (a1 ,a2 )`

v(k) − F n+1 v(k − (n + 1)`) − a2 F n v(k − (n + 2)`) =

n X

F i u(k − i`).

(5)

i=0

Proof. From (1) and the given relation

∆

(a1 ,a2 )`

v(k) = u(k), we arrive

v(k) = u(k) + a1 v(k − `) + a2 v(k − 2`).

(6)

Replacing k by k − ` in (6) and then substituting the value of v(k − `) in (6), we get v(k) = u(k) + a1 u(k − `) + (a21 + a2 )v(k − 2`) + a1 a2 v(k − 3`).

(7)

Using F0 , F1 and F2 given in (2),(7) can be expressed as v(k) = F0 u(k) + F1 u(k − `) + F2 v(k − 2`) + a2 F1 v(k − 3`).

(8)

Replacing k by k − 2` in (6), substituting v(k − 2`) in (8) and (2), we obtain v(k) = F0 u(k) + F1 u(k − `) + F2 u(k − 2`) + F3 v(k − 3`) + a2 F2 v(k − 4`), Now (5) follows by repeating the above processes. Taking u(k) = a sk in (5) we get the following Corollary. a a1 − 2 . Then we have a s` a2s` n  a1 a2  X a sk − F n+1 a s(k−(n+1)`) − a2 F n a s(k−(n+2)`) = 1 − s` − 2s` F i a s(k−i`) . a a

Corollary 2.4. Let k ∈ (−∞, ∞), F n ∈ F(a1 ,a2 ) and 1 ≠

(9)

i=0

In particular, we can replace a by e for exponential function. The following example is an verification of (9). Example 2.5. Taking k = 7, ` = 3, a = 5, a1 = 5, a2 = 7, n = 3, s = 1 and F n ∈ F(5,7) in (9), we get 3  7  P 5 − F4 5−5 − 7F3 5−8 = 0.959552 F i 5(7−3i) = 0.959552 × 81417.8. i=0

Theorem 2.6. If 1 − a1 − a2 ≠ 0, then v(k) − a1 v(k − `) − a2 v(k − 2`) = k m has a closed form solution 1 and 1 − a1 − a2 −1

∆

(a1 ,a2 )`

∆

(a1 ,a2 )`

k0 =

m

km =

X −1 (−1)i mC `i (a1 + 2i a2 )k m−i km i + , ∆ 1 − a1 − a2 1 − a1 − a2 (a1 ,a2 )`

m ≥ 1.

(10)

i=1

Proof. Taking v(k) = k0 in (1) and using (3), we get Taking v(k) =

−1

−1

∆

(a1 ,a2 )`

k0 =

1 . 1 − a1 − a2

k in (1), we get 1 − a1 − a2 k `(a1 + 2a2 )k0 =k+ . (1 − a1 − a2 ) (a1 ,a2 )` (1 − a 1 − a 2 ) ∆

By taking v(k) =

(11)

k2 k3 km , and , in (1) we get 1 − a1 − a2 1 − a1 − a2 1 − a1 − a2 k2 2`(a1 + 2a2 )k `2 (a1 + 4a2 )k0 = k2 + − . (1 − a1 − a2 ) (1 − a1 − a2 ) (a1 ,a2 )` (1 − a 1 − a 2 ) ∆

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(12)

Laplace - Fibonacci transform by the solution of second order generalized difference equation |

3`(a1 + 2a2 )k2 3`2 (a1 + 4a2 )k `3 (a1 + 8a2 )k0 k3 = k3 + − + (1 − a1 − a2 ) (1 − a1 − a2 ) (1 − a1 − a2 ) (a1 ,a2 )` (1 − a 1 − a 2 ) ∆

= k3 +

25

(13)

3 (−1)i+1 3C `i (a + 2i a )k 3−i P 1 2 i . 1 − a1 − a2 i=1

and in general, we obtain m (−1)i+1 mC `i (a + 2i a )k m−i P km 1 2 i . = km + 1 − a1 − a2 (a1 ,a2 )` (1 − a 1 − a 2 ) i=1

∆

The proof of (10) follows from linear property and (3). Substituting

−1

∆

(a1 ,a2 )`

k0 ,

−1

∆

(a1 ,a2 )`

k and

−1

∆

(a1 ,a2 )`

Corollary 2.7. The difference equation

k2 in (10), we have the following corollary.

∆

(a1 ,a2 )` −1

∆

(a1 ,a2 )`

k3 =

v(k) = k3 has a closed form solution

k3 3`(a1 + 2a2 )k2 6`2 (a1 + 2a2 )2 k 3`2 (a1 + 4a2 )k + + − 1 − a1 − a2 (1 − a1 − a2 )2 (1 − a1 − a2 )3 (1 − a1 − a2 )2 −

6`3 (a1 + 2a2 )3 6`3 (a1 + 2a2 )(a1 + 4a2 ) `3 (a1 + 8a2 ) − − . (1 − a1 − a2 )4 (1 − a1 − a2 )3 (1 − a1 − a2 )2

Corollary 2.8. Let v(k) =

−1

∆

(a1 ,a2 )`

(14)

k m be given in (10) and F n ∈ F(a1 ,a2 ) . Then

v(k) − F n+1 v(k − (n + 1)`) − a2 F n v(k − (n + 2)`) =

n X

F i (k − i`)m .

(15)

i=0

Proof. The proof follows by taking u(k) = k m in Theorem 2.3. The following example is a verification of (15). Example 2.9. Take k = 6, ` = 3, m = 3, n = 4, a1 = 4, a2 = 5 and F n ∈ F(4,5) in Corollary (2.8). Then 4 P F i (6 − 3i)3 = v(6) − F5 v(−9) − 5F4 v(−12) = −115020. i=0

Lemma 2.10. [12] Let s nr ’s be the Stirling numbers of the first kind and let m−1 Q k(m) (k − i`) be generalized polynomial factorial. Then we have the relation ` = i=0

k(n) ` =

n X

s nr `n−r k r .

(16)

r=1

Theorem 2.11. For m ≥ 1, a closed form solution of the second order generalized difference equation v(k) − a1 v(k − `) − a2 v(k − 2`) = k(m) ` is given by −1

∆

(a1 ,a2 )`

Proof. Taking

−1

∆

(a1 ,a2 )`

k(m) ` =

m X r m m−r+i m−r r X X −1 (−1)i rC i s m (a1 + 2i a2 )k r−i sm k r ` r ` + . ∆ 1 − a1 − a2 1 − a1 − a2 (a1 ,a2 )` r=1

(17)

r=1 i=1

on (16) and applying Theorem 2.6 to k r , we get (17).

Corollary 2.12. A closed form solution of the equation

∆

(a1 ,a2 )`

v(k) = k(2) ` is

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26 | Sandra Pinelas et al. −1

∆

(a1 ,a2 )`

k(2) ` =

k2 `k `2 (a1 + 2a2 ) 2`(a1 + 2a2 )k − + − 1 − a1 − a2 1 − a1 − a2 (1 − a1 − a2 )2 (1 − a1 − a2 )2 +

`2 (a1 + 4a2 ) 2`2 (a1 + 2a2 )2 + . 3 (1 − a1 − a2 ) (1 − a1 − a2 )2

(18)

Proof. Putting m = 2 in (17) and since s21 = −1, s22 = 1, we find that −1

∆

(a1 ,a2 )`

k(2) ` =

`k `2 (a1 + 2a2 ) −1 k2 0 − + ∆ k 1 − a1 − a2 1 − a1 − a2 (1 − a1 − a2 ) (a1 ,a2 )`

− Since

−1

∆

(a1 ,a2 )`

k=

`2 (a1 + 4a2 ) −1 2`(a1 + 2a2 ) −1 0 ∆ k+ ∆ k . (1 − a1 − a2 ) (a1 ,a2 )` (1 − a1 − a2 ) (a1 ,a2 )`

−1 k 1 `(a1 + 2a2 ) , ∆ k0 = − , we get (18). 1 − a1 − a2 (1 − a1 − a2 )2 (a1 ,a2 )` 1 − a1 − a2

Corollary 2.13. If v(k) =

−1

∆

(a1 ,a2 )`

k(m) ` is as given in (18), then we have

v(k) − F n+1 v(k − (n + 1)`) − a2 F n v(k − (n + 2)`) =

n X

F i (k − i`)(m) ` .

(19)

i=0

Proof. The proof follows by taking u(k) = k(m) ` in Theorem 2.3. Example 2.14. Take k = 12, ` = 4, m = 2, n = 3, a1 = 7, a2 = 10 and F n ∈ F(7,10) in Corollary (2.13). Then 3 P F i (12 − 4i)(2) 4 = v(12) − F 4 v(−4) − 10F 3 v(−8) = 320. i=0

Following theorem gives the inverse of

∆

(a1 ,a2 )`

on product of two functions.

Theorem 2.15. Let u(k) and v(k) be two real valued functions. Then i i −1 h −1 −1 h −1 u(k)v(k) = u(k) ∆ v(k) − a1 ∆ ∆ ∆ v(k − `) ∆ u(k)

(a1 ,a2 )`

(a1 ,a2 )`

(a1 ,a2 )`

−a2

−1

∆

h

(a1 ,a2 )`

(a1 ,a2 )` −1

∆

(a1 ,a2 )`

(1,0)`

i v(k − 2`) ∆ u(k) .

(20)

(0,1)`

Proof. hFrom the idefinition of the difference operator given in (1), we arrive u(k)w(k) = u(k)w(k) − a1 u(k − `)w(k − `) − a2 u(k − 2`)w(k − 2`). ∆ (a1 ,a2 )`

Adding and subtracting a1 u(k)w(k − `), a2 u(k)w(k − 2`) on the right side, we obtain h i u(k)w(k) = u(k) ∆ w(k) + a1 w(k − `) ∆ u(k) + a2 w(k − 2`) ∆ u(k). ∆

(a1 ,a2 )`

(a1 ,a2 )`

Taking w(k) =

−1

∆

(a1 ,a2 )`

v(k) and applying

(1,0)`

−1

∆

(a1 ,a2 )`

(0,1)`

on both sides, we get (20).

Corollary 2.16. A closed form solution of the second order generalized difference equation v(k) − a1 v(k − `) − a2 v(k − 2`) = k2 a k is given by −1

∆

(a1 ,a2 )`

k2 a k =

+

k2 a k 2a1 `ka k−` 4a2 `ka k−2` − −
a a2 a a2 2 a a 2
2 1 − 1` − 2` 1 − 1` − 2` 1 − 1` − 2` a a a a a a 2a21 `2 a k−2` a1 `2 a k−` 4a2 `2 a k−2` a 2
2 + a 2
2 + a 2
3 a1 a1 a 1 − ` − 2` 1 − ` − 2` 1 − 1` − 2` a a a a a a

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Laplace - Fibonacci transform by the solution of second order generalized difference equation |

8a22 `2 a k−4` 8a1 a2 `2 a k−3` a1 a 2
3 + a a 2
3 . 1 − ` − 2` 1 − 1` − 2` a a a a

+

27

(21)

Proof. Taking u(k) = k and v(k) = a k in (20) and using (4), (11), we arrive −1

∆

(a1 ,a2 )`

ka k =

a1 `a k−` 2a2 `a k−2` ka k a2 − a1 a 2
2 − a a 2
2 . a1 1 − ` − 2` 1 − 1` − 2` 1 − ` − 2` a a a a a a

(22)

Again, taking u(k) = k2 and v(k) = a k in (20) and using (4), (12), we arrive    k−` 2 k −1 −1 a k a 2 k  (2k − `)` ∆  ∆ k a = a2 − a1 (a ,a a2 a a (a1 ,a2 )` 1 2 )` 1 − 1` − 2` 1 − 1` − 2` a a a a    k−2` −1 a  (4k − 4`)`. −a2 ∆  a2 a (a1 ,a2 )` 1 − 1` − 2` a a Substituting (22) in the above and using (4), we get the proof of (21). Corollary 2.17. If v(k) =

−1

∆

(a1 ,a2 )`

k2 a k is the solution given in (21), then we have

v(k) − F n+1 v(k − (n + 1)`) − a2 F n v(k − (n + 2)`) =

n X

F i (k − i`)2 a k−i` .

(23)

i=0

Proof. The proof follows by taking u(k) = k2 a k in Theorem 2.3. Example 2.18. Take k = 7, ` = 2, n = 3, a = 5, a1 = 10 and a2 = 12 in Corollary (2.17). Then

3 P

F i (7 −

i=0

2i)2 57−2i = v(7) − F4 v(−1) − 12F3 v(−3) = 4741575. Corollary 2.19. A closed form solution of the second order generalized difference equation v(k) − a1 v(k − `) − a2 v(k − 2`) = ke−sk is given by −1

∆

(a1 ,a2 )`

ke−sk =

a1 `e−s(k−`) 2a2 `e−s(k−2`) ke−sk − − . 1 − a1 e s` − a2 e2s` (1 − a1 e s` − a2 e2s` )2 (1 − a1 e s` − a2 e2s` )2

(24)

and hence we arrive n X

v(k) − F n+1 v(k − (n + 1)`) − a2 F n v(k − (n + 2)`) =

F i (k − i`)e−s(k−i`) .

(25)

i=0

Proof. Taking a = e−1 in (22), we get (24). Now (25) follows by Theorem 2.3. Example 2.20. Take k = 5, ` = 0.8, n = 4, a1 = 3 and a2 = 5 in (25). Then 4 F (5 − 0.8i) P i = v(5) − F5 v(1) − 5F4 v(0.2) = 84.52520. e(5−0.8i) i=0

3 Laplace-Fibonacci Transforms and its Applications The operators ∆−1 t and form (

−1

∆

−1

∆

(a1 ,a2 )`

motivate us to define the discrete Laplace Transform (∆−1 t ), Fibonacci Trans-

) and discrete Laplace-Fibonacci transform (∆−1 t {

(a1 ,a2 )`

−1

∆

}). For that we replace single integral by

(a1 ,a2 )`

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28 | Sandra Pinelas et al. ∆−1 t and double integral by

−1

∆

(a1 ,a2 )`

in the Laplace transform of Digital signal processing and Digital image

processing. Thus in this section, we obtain Laplace-Fibonacci transform for certain functions (input signals) and with the help of MATLAB we analyze the above transform generated by the inverse of two dimensional difference operator. Definition 3.1. Let t > 0, u(k) be be defined for k ≥ 0. Then for (a1 , a2 ) ∈ R2 , the Discrete Laplace and Laplace-Fibonacci transforms are respectively defined as −sk ∞ L t [u(k)] = t∆−1 t u(k)e k=0 and LF t [u(k)] = t∆−1 t {

−1

∆

(a1 ,a2 )`

In the following example we discuss the outcomes of

∞ u(k)}e−sk k=0

−1

∆

(26)

(a1 ,a2 )`

(27)

on k2 a k .

 a a2 −1 Example 3.2. By taking A = 1 − 1` − 2` , equation (21) becomes a a −1

∆

(a1 ,a2 )`

where B =

−

2a1 4a2
+ 2` ` a` a and C = A2

k2 a k = Ak2 a k + Bka k + Ca k ,

(28)

2a21 8a1 a2 8a22
2 a1 4a2
2 + 2` ` + + 4` ` ` a a a2` a3` a + . A2 A3

Taking a = 5, a1 = 10 and a2 = 0 in 28, we find the following outcomes.

From the outcomes, we observed that there is a rapid change when ` > 1.43. Theorem 3.3. If 1 − a t e−st ≠ 0, s ≥ 0 and A ≠ 0, then we have LF t [k2 a k ] = −

2At3 a2t e−2st At3 a t e−st Bt2 a t e−st Ct + t −st + t −st − t −st 3 2 (a e − 1) (a e − 1) (a e − 1)2 a t e−st − 1

where A, B and C are defined as in Example 3.2. Proof. Taking u(k) = a k , ka k and k2 a k in (27), we find

Unauthenticated Download Date | 9/2/17 2:39 PM

(29)

Laplace - Fibonacci transform by the solution of second order generalized difference equation |

L t [a k ] = −

1

and L t [ka k ] =

a t e−st − 1 2 2t −2st

29

ta t e−st − 1)2

(a t e−st 2 t −st

2t a e t ae + . (a t e−st − 1)3 (a t e−st − 1)2 Now the proof follows by applying (27) in (28) and using the above results. L t [k2 a k ] = −

In the following example, we discuss the LF t (k2 a k ) Example 3.4. Taking a = 5, a1 = 10, a2 = 0 and t = 1.5 in Theorem 3.3, we have following outcomes

Due to the rapid change of

−1

∆

(10,0)`

k2 5k in Example 3.2, we have obtained a corresponding change in the

Fibonacci-Laplace transform with respect to t > 0. By tuning the value of ’t’ we can select optimal outcome based on our need.

4 Conclusion In this paper, we have obtained several formulae on two dimensional Fibonacci series by equating closed form and summation form solutions of the generalized difference equation using ∆−1 t or

−1

∆

(a1 ,a2 )`

or both. From

the diagram given in example (3.4), we have observed that there are rapid changes when we apply LaplaceFibonacci transform to the input signal k2 a k . Replacing Laplace transform by Laplace-Fibonacci transform, we have the possibility of several applications in Digital Signal Processing and Image Processing. We have also discussed discrete Laplace-Fibonacci transform for the functions (signals) of arithmetic form and exponential form. In the example (3.2), discrete Fibonacci transform makes a rapid changes for arithmetic-geometric function. Finally, discrete Laplace transform gives uncountably many outcomes as t varies on (0, ∞). From our findings we suggest that by interchanging integral by ∆−1 t , the readers (researchers) can obtain innumerable applications in Digital Signal processing and Image Processing. Our future research will progress in this direction.

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30 | Sandra Pinelas et al.

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