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

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1 Nonauton. Dyn. Syst. 017; 4: 30 Research Artice Open Access Sandra Pineas*, G.B.A Xavier, S.U. Vasantha Kumar, and M. Meganathan Lapace - Fibonacci transform by the soution of second order generaized difference equation Received June 9, 016; accepted Apri 4, 017 Abstract: The main objective of this paper is finding new types of discrete transforms with tuning factor t. This is not ony anaogy to the continuous Lapace transform but gives discrete Lapace-Fibonacci transform (LF t ). This type of Lapace-Fibonacci transform is not avaiabe in the continuous case. The LF t generates uncountaby many outcomes when the parameter t varies on (0, ). This possibiity is not avaiabe in the existing Lapace transform. A the formuae and resuts derived are verified by MATLAB. Keywords: Generaized difference operator, Two dimensiona Fibonacci sequence, Cosed form soution, Fibonacci summation formua, Lapace-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 Facon and Paza [3] depends ony on one integer parameter k and is defined as F k,0 0, F k,1 1 and F k,n1 kf k,n F k,n, where n 1, k 1. In particuar, if k, the Pe sequence is obtained as P 0 0, P 1 1 and P n1 P n P n for n 1. In 015, M. Lawrence Gasser and Yajun Zhou [9] report on an Integra representation for the Fibonacci Numbers and their Generaization. In 015, Martin Griffiths and Wiiam Wynn-Thomas [11] construct, by way of the Fibonacci numbers, a geometrica object resembing an infinite staircase and demonstrate an interesting property of this Fibonacci staircase. Koshy [8] construct graph-theoretic modes for an extended univariate Fibonacci famiy, which incudes Fibonacci, Lucas, Pe, and Pe-Lucas poynomias. In 016, Jeremy F. Am and Tayor Herad [5] define a variant of Fibonacci-ike sequences(prime Fibonacci sequences), where one takes the sum of the previous two terms and returns the smaest odd prime divisor of that sum as the next term and proved some resuts. Mahadi Ddamuira, Forian Luca and Mihaja Rakotomaaa [10] find a Fibonacci numbers which are products of two Pe numbers and a Pe numbers which are products of two Fibonacci numbers. R.S.Meham (refer [14], [15]) find cosed forms, in terms of rationa numbers, for certain finite sums. His most genera resuts are for finite sums where the denominator of the summand is a product of terms from a se- *Corresponding Author: Sandra Pineas: Academia Miitar, Departamento de CiÃłncias Exactas e Naturais, Av. Conde Castro GuimarÃčes, Amadora, Portuga, E-mai: sandra.pineas@gmai.com G.B.A Xavier: Department of Mathematics, Sacred Heart Coege, Tirupattur , Veore District, Tami Nadu, S. India, E-mai: brittoshc@gmai.com S.U. Vasantha Kumar, M. Meganathan: Department of Mathematics, Sacred Heart Coege, Tirupattur , Veore District, Tami Nadu, S. India Open Access. Âľ 017 Sandra Pineas et a., pubished by De Gruyter Open. This work is icensed under the Creative Commons Attribution-NonCommercia-NoDerivs 3.0 License. Downoad Date 9/18/18 9:10 PM

2 Lapace - Fibonacci transform by the soution of second order generaized difference equation 3 quence that generaizes both the Fibonacci and Lucas numbers. Aso he created a ink to the Fibonacci/Lucas numbers then faciitates the derivation of cosed forms for reciproca series that invove the Fibonacci/Lucas numbers. The term that defines the denominator of each summand contains squares of Fibonacci reated numbers, with subscripts in arithmetic progression. To deveop Two Dimensiona Fibonacci sequence and Lapace-Fibonacci transform, we need to revea basic theory of generaized difference operators and α(). In 1984, Jerzy Popenda [6] introduced a particuar type of difference operator on u(k) as α u(k) u(k 1) αu(k), In 011, M.Maria Susai Manue, et.a, [13] extended the operator α to generaized α difference operator as v(k) v(k ) αv(k) for the rea α() vaued function v(k). These operators induces to introduce the foowing second order generaized difference operator. Here, for each pair (, a ) R, where R is the set of a rea numbers and > 0, second order generaized difference operator on v(k) is defined as v(k) v(k) v(k ) a v(k ), k (, ), (1) which generates two dimensiona Fibonacci sequence and its sum. By taking α, a 0 in (1) we can get the above two operators. For (, a ) R, a two dimensiona Fibonacci sequence is defined as F (a1,a ) {F n } n0, where F 0 1, F 1, F n F n a F n, n. () The sequence () becomes the we known Fibonacci sequence when a 1. For exampe, the two dimensiona Fibonacci sequences F (8,16) {1, 8, 80, 768, } and F (0.7,1.5) {1, 0.7, 1.99,.443, , } are obtained from () by taking 8, a 16 and 0.7, a 1.5 respectivey. The Pe sequence F (,1) {0, 1,, 5, 1, 9, 70,...} is obtained by taking F 0 0, F1 1 and F n F n F n for n. Simiary, one can obtain two dimensiona Fibonacci sequence corresponding to each pair (, a ) R. Sum of Fibonacci Numbers With Function Vaues The operator defined in (1) generates severa formuae directy on Fibonacci numbers. The method to derive the above is simpe. First we need to form a generaized difference equation and then by equating the cosed form and summation soutions, we get the desired formua. Hence in this section, we derive some formuae on sum of Fibonacci sequence using inverse of second order difference operator. Definition.1. For each pair (, a ) R and (0, ), consider (1). If v(k) u(k), then we write v(k) u(k). (3) Exampe.. Taking u(k) a sk and v(k) a sk [ 1 a s a a s ] in (3), we get a sk a sk 1 a s a a s, where Hereafter, we assume that F n F (a1,a ) and 1 a 0. 1 a s a 0. (4) as Downoad Date 9/18/18 9:10 PM

3 4 Sandra Pineas et a. Theorem.3. (Fibonacci-Function vaue Summation Formua) Let F n F (a1,a ) and v(k) be a soution to the second order generaized difference equation v(k) u(k), k (, ). Then we have Proof. From (1) and the given reation v(k) F n1 v(k (n 1)) a F n v(k (n )) v(k) u(k), we arrive F i u(k i). (5) v(k) u(k) v(k ) a v(k ). (6) Repacing k by k in (6) and then substituting the vaue of v(k ) in (6), we get v(k) u(k) u(k ) (a 1 a )v(k ) a v(k 3). (7) Using F 0, F 1 and F given in (),(7) can be expressed as v(k) F 0 u(k) F 1 u(k ) F v(k ) a F 1 v(k 3). (8) Repacing k by k in (6), substituting v(k ) in (8) and (), we obtain v(k) F 0 u(k) F 1 u(k ) F u(k ) F 3 v(k 3) a F v(k 4), Now (5) foows by repeating the above processes. Taking u(k) a sk in (5) we get the foowing Coroary. Coroary.4. Let k (, ), F n F (a1,a ) and 1 a s a. Then we have as ( a sk F n1 a s(k (n1)) a F n a s(k (n)) 1 a s a ) a s F i a s(k i). (9) In particuar, we can repace a by e for exponentia function. The foowing exampe is an verification of (9). Exampe.5. Taking k 7, 3, a 5, 5, a 7, n 3, s 1 and F n F (5,7) in (9), we get [ 5 7 F F 3 5 8] F i 5 (7 3i) Theorem.6. If 0, then v(k) v(k ) a v(k ) k m has a cosed form soution 1 and k m k m Proof. Taking v(k) k 0 in (1) and using (3), we get Taking v(k) By taking v(k) k in (1), we get k, m i1 k 3 and k 0 () i mc i i ( i a )k m i, m 1. (10) k 0 1. k ( ) k ( a )k 0 ( ). (11) k m, in (1) we get k ( ) k ( a )k ( ) ( 4a )k 0 ( ). (1) Downoad Date 9/18/18 9:10 PM

4 Lapace - Fibonacci transform by the soution of second order generaized difference equation 5 and in genera, we obtain k 3 ( ) k3 3( a )k 3 ( 4a )k 3 ( 8a )k 0 ( ) ( ) ( ) k 3 3 i1 k m ( ) km m () i1 3C i i ( i a )k 3 i. i1 () i1 mc i i ( i a )k m i. The proof of (10) foows from inear property and (3). Substituting k 0, k and k in (10), we have the foowing coroary. (13) Coroary.7. The difference equation v(k) k 3 has a cosed form soution k 3 k 3 3( a )k ( ) 6 ( a ) k ( ) 3 3 ( 4a )k ( ) Coroary.8. Let v(k) 63 ( a ) 3 ( ) 4 63 ( a )( 4a ) ( ) 3 3 ( 8a ) ( ). (14) k m be given in (10) and F n F (a1,a ). Then v(k) F n1 v(k (n 1)) a F n v(k (n )) F i (k i) m. (15) Proof. The proof foows by taking u(k) k m in Theorem.3. The foowing exampe is a verification of (15). Exampe.9. Take k 6, 3, m 3, n 4, 4, a 5 and F n F (4,5) in Coroary (.8). Then 4 F i (6 3i) 3 v(6) F 5 v( 9) 5F 4 v() Lemma.10. [1] Let s n r s be the Stiring numbers of the first kind and et m (k i) be generaized poynomia factoria. Then we have the reation k (m) k (n) s n r n r k r. (16) r1 Theorem.11. For m 1, a cosed form soution of the second order generaized difference equation v(k) v(k ) a v(k ) k (m) is given by Proof. Taking k (m) m r1 s m r m r k r m r1 r i1 on (16) and appying Theorem.6 to k r, we get (17). Coroary.1. A cosed form soution of the equation () i rc i s m r m ri ( i a )k r i. (17) v(k) k () is Downoad Date 9/18/18 9:10 PM

5 6 Sandra Pineas et a. k () k k ( a ) ( ) ( a )k ( ) Proof. Putting m in (17) and since s 1, s 1, we find that k () Since k ( a ) ( ) ( a ) ( ) 3 ( 4a ) ( ). (18) k ( a ) ( ) k ( 4a ) ( ) k k ( a ) ( ), Coroary.13. If v(k) k 0 k 0. k 0 k (m) is as given in (18), then we have 1, we get (18). v(k) F n1 v(k (n 1)) a F n v(k (n )) F i (k i) (m). (19) Proof. The proof foows by taking u(k) k (m) in Theorem.3. Exampe.14. Take k 1, 4, m, n 3, 7, a 10 and F n F (7,10) in Coroary (.13). Then 3 F i (1 4i) () 4 v(1) F 4v( 4) 10F 3 v( 8) 30. Foowing theorem gives the inverse of on product of two functions. Theorem.15. Let u(k) and v(k) be two rea vaued functions. Then [ ] [ u(k)v(k) u(k) v(k) v(k ) a [ v(k ) (1,0) (0,1) ] u(k) Proof. [ From the ] definition of the difference operator given in (1), we arrive u(k)w(k) u(k)w(k) u(k )w(k ) a u(k )w(k ). Adding and subtracting u(k)w(k ), a u(k)w(k ) on the right side, we obtain [ ] u(k)w(k) u(k) w(k) w(k ) (1,0) u(k) a w(k ) (0,1) u(k). Taking w(k) v(k) and appying on both sides, we get (0). ] u(k). (0) Coroary.16. A cosed form soution of the second order generaized difference equation v(k) v(k ) a v(k ) k a k is given by k a k k a k 1 a a a a k a a ) a ka k ( 1 a a a ) 4a a k a a ) a 4a ka k a a ) a a 1 a k a a ) 3 a Downoad Date 9/18/18 9:10 PM

6 Lapace - Fibonacci transform by the soution of second order generaized difference equation 7 8a a k 3 8a a a ) 3 a k 4 a a a ) 3. (1) a Proof. Taking u(k) k and v(k) a k in (0) and using (4), (11), we arrive ka k ka k 1 a a a a k ( 1 a a a ) Again, taking u(k) k and v(k) a k in (0) and using (4), (1), we arrive k a k k a k 1 a a a k a 1 a a (k ) a a a k 1 a a a (4k 4). Substituting () in the above and using (4), we get the proof of (1). Coroary.17. If v(k) k a k is the soution given in (1), then we have v(k) F n1 v(k (n 1)) a F n v(k (n )) Proof. The proof foows by taking u(k) k a k in Theorem.3. a a k ( 1 a a a ). () F i (k i) a k i. (3) Exampe.18. Take k 7,, n 3, a 5, 10 and a 1 in Coroary (.17). Then 3 F i (7 i) 5 7 i v(7) F 4 v() 1F 3 v( 3) Coroary.19. A cosed form soution of the second order generaized difference equation v(k) v(k ) a v(k ) ke sk is given by and hence we arrive ke sk ke sk 1 e s a e s e s(k ) (1 e s a e s ) a e s(k ) (1 e s a e s ). (4) v(k) F n1 v(k (n 1)) a F n v(k (n )) Proof. Taking a e in (), we get (4). Now (5) foows by Theorem.3. Exampe.0. Take k 5, 0.8, n 4, 3 and a 5 in (5). Then 4 F i (5 0.8i) e (5 0.8i) v(5) F 5 v(1) 5F 4 v(0.) F i (k i)e s(k i). (5) 3 Lapace-Fibonacci Transforms and its Appications The operators t motivate us to define the discrete Lapace Transform ( t ), Fibonacci Transform ( and ) and discrete Lapace-Fibonacci transform ( t { }). For that we repace singe integra by Downoad Date 9/18/18 9:10 PM

7 8 Sandra Pineas et a. t and doube integra by in the Lapace transform of Digita signa processing and Digita image processing. Thus in this section, we obtain Lapace-Fibonacci transform for certain functions (input signas) and with the hep of MATLAB we anayze the above transform generated by the inverse of two dimensiona difference operator. Definition 3.1. Let t > 0, u(k) be be defined for k 0. Then for (, a ) R, the Discrete Lapace and Lapace-Fibonacci transforms are respectivey defined as and L t [u(k)] t t u(k)e sk k0 LF t [u(k)] t t { u(k)}e sk k0 (6) (7) In the foowing exampe we discuss the outcomes of Exampe 3.. By taking A where B ( a 4a ) ( a 1 a A and C on k a k. ( 1 a a a ), equation (1) becomes k a k Ak a k Bka k Ca k, (8) a 4a a ) A ( a 1 a 8a a 3 8a a 4 ) A 3. Taking a 5, 10 and a 0 in 8, we find the foowing outcomes. From the outcomes, we observed that there is a rapid change when > Theorem 3.3. If 1 a t e st 0, s 0 and A 0, then we have LF t [k a k ] At3 a t e st (a t e st 1) 3 At3 a t e st (a t e st 1) Bt a t e st (a t e st 1) Ct a t e st 1 (9) where A, B and C are defined as in Exampe 3.. Proof. Taking u(k) a k, ka k and k a k in (7), we find Downoad Date 9/18/18 9:10 PM

8 Lapace - Fibonacci transform by the soution of second order generaized difference equation 9 L t [a k 1 ] a t e st 1 and L t[ka k ta t e st ] (a t e st 1) L t [k a k ] t a t e st (a t e st 1) 3 t a t e st (a t e st 1). Now the proof foows by appying (7) in (8) and using the above resuts. In the foowing exampe, we discuss the LF t (k a k ) Exampe 3.4. Taking a 5, 10, a 0 and t 1.5 in Theorem 3.3, we have foowing outcomes Due to the rapid change of (10,0) k 5 k in Exampe 3., we have obtained a corresponding change in the Fibonacci-Lapace transform with respect to t > 0. By tuning the vaue of t we can seect optima outcome based on our need. 4 Concusion In this paper, we have obtained severa formuae on two dimensiona Fibonacci series by equating cosed form and summation form soutions of the generaized difference equation using t or or both. From the diagram given in exampe (3.4), we have observed that there are rapid changes when we appy Lapace- Fibonacci transform to the input signa k a k. Repacing Lapace transform by Lapace-Fibonacci transform, we have the possibiity of severa appications in Digita Signa Processing and Image Processing. We have aso discussed discrete Lapace-Fibonacci transform for the functions (signas) of arithmetic form and exponentia form. In the exampe (3.), discrete Fibonacci transform makes a rapid changes for arithmetic-geometric function. Finay, discrete Lapace transform gives uncountaby many outcomes as t varies on (0, ). From our findings we suggest that by interchanging integra by t, the readers (researchers) can obtain innumerabe appications in Digita Signa processing and Image Processing. Our future research wi progress in this direction. Downoad Date 9/18/18 9:10 PM

9 30 Sandra Pineas et a. References [1] Bastos.N. R. O, Ferreira.R. A. C, and Torres.D. F. M. Discrete-Time Fractiona Variationa Probems, Signa Processing, 91(3)(011), [] Britto Antony Xavier.G, Gery.T.G and Nasira Begum.H, Finite Series of Poynomias and Poynomia Factorias arising from Generaized q-difference operator, Far East Journa of Mathematica Sciences,94(1)(014), [3] Facon.S and Paza.A, "On the Fibonacci k numbers", Chaos, Soitons and Fractas, vo.3, no.5 (007), pp [4] Ferreira.R. A. C and Torres.D. F. M, Fractiona h-difference equations arising from the cacuus of variations, Appicabe Anaysis and Discrete Mathematics, 5(1) (011), [5] Jeremy F. Am and Tayor Herad, A Note on Prime Fibonacci Sequences, Fibonacci Quartery, 54 (016), no. 1, [6] Jerzy Popenda and Bazej Szmanda, On the Osciation of Soutions of Certain Difference Equations, Demonstratio Mathematica, XVII(1), (1984), [7] Koshy.T, Fibonacci and Lucas Numbers with Appications, Wiey-Interscience, New York, NY, USA, 001. [8] Koshy.T, Graph-Theoretic Modes for the Univariate Fibonacci Famiy, Fibonacci Quartery, 53 (015), no., [9] M. Lawrence Gasser and Yajun Zhou, An Integra Representation for the Fibonacci Numbers and Their Generaization, Fibonacci Quartery, 53 (015), no. 4, [10] Mahadi Ddamuira, Forian Luca and Mihaja Rakotomaaa, Fibonacci Numbers Which are Products of Two Pe Numbers, Fibonacci Quartery, 54 (016), no. 1, [11] Martin Griffiths and Wiiam Wynn-Thomas, A Property of a Fibonacci Staircase, Fibonacci Quartery, 53 (015), no. 1, [1] Maria Susai Manue.M, Britto Antony Xavier.G and Thandapani.E, Theory of Generaized Difference Operator and Its Appications, Far East Journa of Mathematica Sciences, 0() (006), [13] Maria Susai Manue.M, Chandrasekar.V and Britto Antony Xavier.G, Soutions and Appications of Certain Cass of α- Difference Equations, Internationa Journa of Appied Mathematics, 4(6) (011), [14] Meham.R.S, On Certain Famiies of Finite Reciproca Sums that Invove Generaized Fibonacci Numbers, Fibonacci Quartery, 53 (015), no. 4, [15] Meham.R.S, More New Agebraic Identities and the Fibonacci Summations Derived From Them, Fibonacci Quartery, 54 (016), no. 1, [16] Mier.K.S and Ross.B, Fractiona Difference Cacuus in Univaent Functions, Horwood, Chichester, UK, (1989), [17] Susai Manue.M, Britto Antony Xavier.G, Chandrasekar.V and Pugaarasu.R, Theory and appication of the Generaized Difference Operator of the n th kind(part I), Demonstratio Mathematica, 45(1)(01), [18] Vajda.S, Fibonacci and Lucas Numbers, and the Goden Section, Eis Horwood, Chichester, UK, Downoad Date 9/18/18 9:10 PM