Research Article On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

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1 Hidawi Publishig Corporatio Iteratioal Joural of Mathematics ad Mathematical Scieces Volume 009, Article ID , 1 pages doi: /009/ Research Article O Sequeces of Numbers ad Polyomials Defied by Liear Recurrece Relatios of Order Tia-Xiao He 1 ad Peter J.-S. Shiue 1 Departmet of Mathematics ad Computer Sciece, Illiois Wesleya Uiversity, Bloomigto, IL 6170, USA Departmet of Mathematical Scieces, Uiversity of Nevada Las Vegas, Las Vegas, NV 89154, USA Correspodece should be addressed to Tia-Xiao He, the@iwu.edu Received 7 July 009; Accepted 14 September 009 Recommeded by Misha Rudev Here we preset a ew method to costruct the explicit formula of a sequece of umbers ad polyomials geerated by a liear recurrece relatio of order. The applicatios of the method to the Fiboacci ad Lucas umbers, Chebyshev polyomials, the geeralized Gegebauer-Humbert polyomials are also discussed. The derived idea provides a geeral method to costruct idetities of umber or polyomial sequeces defied by liear recurrece relatios. The applicatios usig the method to solve some algebraic ad ordiary differetial equatios are preseted. Copyright q 009 T.-X. He ad P. J.-S. Shiue. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. 1. Itroductio May umber ad polyomial sequeces ca be defied, characterized, evaluated, ad classified by liear recurrece relatios with certai orders. A umber sequece {a } is called sequece of order if it satisfies the liear recurrece relatio of order : a pa 1 qa,, 1.1 for some ozero costats p ad q ad iitial coditios a 0 ad a 1. I Masour 1, the sequece {a } 0 defied by 1.1 is called Horadam s sequece, which was itroduced i 1965 by Horadam. The work i 1 also obtaied the geeratig fuctios for powers of Horadam s sequece. To costruct a explicit formula of its geeral term, oe may use a geeratig fuctio, characteristic equatio, or a matrix method see Comtet 3, Hsu 4, Strag 5, Wilf 6, etc. I 7, Bejami ad Qui preseted may elegat combiatorial meaigs of the sequece defied by recurrece relatio 1.1. For istace, a couts

2 Iteratioal Joural of Mathematics ad Mathematical Scieces the umber of ways to tile a -board i.e., board of legth with squares represetig 1ss ad domioes represetig s where each tile, except the iitial oe has a color. I additio, there are p colors for squares ad q colors for domioes. I this paper, we will preset a ew method to costruct a explicit formula of {a } geerated by 1.1. Thekey idea of our method is to reduce the relatio 1.1 of order to a liear recurrece relatio of order 1: a ca 1 d, 1, 1. for some costats c / 0add ad iitial coditio a 0 via geometric sequece. The, the expressio of the geeral term of the sequece of order ca be obtaied from the formula of the geeral term of the sequece of order 1: a 0 c d c 1 a c 1, if c / 1; a 0 d, if c The method ad some related results o the geeralized Gegebauer-Humbert polyomial sequece of order as well as a few examples will be give i Sectio. Sectio 3 will discuss the applicatio of the method to the costructio of the idetities of sequeces of order. There is a extesio of the above results to higher order cases. I Sectio 4, we will discuss the applicatios of the method to the solutio of algebraic equatios ad iitial value problems of secod-order ordiary differetial equatios.. Mai Results ad Examples Let α ad β be two roots of of quadratic equatio x px q 0. We may write 1.1 as a α β ) a 1 αβa,.1 where α ad β satisfy α β p ad αβ q. Therefore, from.1, we have a αa 1 β a 1 αa,. which implies that {a αa 1 } 1 is a geometric sequece with commo ratio β. Hece, a αa 1 a 1 αa 0 β 1, a αa 1 a 1 αa 0 β 1..3 Cosequetly, a β α β a 1 β 1 ) a 1 αa 0..4 β

3 Iteratioal Joural of Mathematics ad Mathematical Scieces 3 Let b : a /β. We may write.4 as b α β b 1 a 1 αa 0..5 β If α / β, byusig 1.3, we immediately obtai ) a α β a 0 a ) 1 αa 0 α/β 1 ) β β α/β 1 1 [ β α a 0 a 1 αa 0 α β )], α β.6 which yields a ) a1 βa 0 α α β ) a1 αa 0 β. α β.7 Similarly, if α β, the 1.3 implies a a 0 α α 1 a 1 αa 0 a 1 α 1 1 a 0 α..8 We may summarize the above result as follows. Propositio.1. Let {a } be a sequece of order satisfyig liear recurrece relatio.1. The ) ) a1 βa 0 α a1 αa 0 β, if α / β; α β α β a a 1 α 1 1 a 0 α, if α β..9 I particular, if {a } satisfies the liear recurrece relatio 1.1 with q 1, amely, a pa 1 a,.10 the the equatio x px 1 0 has two solutios: α p p 4, β 1 α p p From Propositio.1, we have the followig corollary.

4 4 Iteratioal Joural of Mathematics ad Mathematical Scieces Corollary.. Let {a } be a sequece of order satisfyig the liear recurrece relatio a pa 1 a.the a a 1 ) p 4 p p 4 a 0 α a 1 ) p 4 p p 4 a 0 1 α),.1 where α is defied by.11. Similarly, let {a } be a sequece of order satisfyig the liear recurrece relatio a a 1 qa.the a 1 1 4q 1 ) a 0 α 1 4q 1 a 1 1 4q 1 ) a 0 α 4q 1, if q / 1 4 ; a 1 a 1 1 a 0, if q 1 4,.13 where α 1 1/ 1 4q 1 ad α 1/ 1 4q 1 are solutios of the equatio x x q 0. The first special case.1 was studied by Falbo i 8. Ifp 1, the sequece is clearly the Fiboacci sequece. If p q 1, the correspodig sequece is the sequece of umerators whe two iitial coditios are 1 ad 3 or deomiators whe two iitial coditios are 1 ad of the coverget of a cotiued fractio to : {1/1, 3/, 7/5, 17/1, 41/9,...}, called the closest ratioal approximatio sequece to. The secod special case is also a corollary of Propositio.1.Ifq p 1, {a } is the Jacobsthal sequece see Bergum et al. 9. Remark.3. Propositio.1 ca be exteded to the liear recurrece relatios of order with more geeral form: a pa 1 qa l for p q / 1. It ca be see that the above recurrece relatio is equivalet to the form 1.1 b pb 1 qb, where b a k ad k l/ 1 p q. We ow show some examples of the applicatios of our method icludig the presetatio of much easier proofs of some well-kow formulas of the sequeces of order. Remark.4. Deote u [ a 1 α ] [ ] p q, A We may write relatio a pa 1 qa ad a 1 a 1 ito a matrix form u 1 Au with respect to matrix A defied above. Thus u 1 A 1 u 0. To fid explicit expressio of u 1, the real problem is to calculate A 1. The key lies i the eigevalues ad eigevectors. The eigevalues of A are precisely α ad β, which are two roots of the characteristic equatio x px q 0 for the matrix A. However, a obvious idetity ca be obtaied from u,u 1 A 1 u 1,u 0 by takig determiats o the both sides: a 1 a 1 a q 1 a a 0 a 1 see, e.g., 5 for more details.

5 Iteratioal Joural of Mathematics ad Mathematical Scieces 5 Example.5. Let {F } 0 be the Fiboacci sequece with the liear recurrece relatio F F 1 F, where F 0 ad F 1 are assumed to be 0 ad 1, respectively. Thus, the recurrece relatio is a special case of 1.1 with p q 1 ad the special case of the sequece i Corollary., which ca be writte as.1 with α 1 5, β Sice α β 5, from.1 we have the expressio of F as follows: F 1 { ) ) } Example.6. We have metioed above that the deomiators of the closest ratioal approximatio to form a sequece satisfyig the recurrece relatio a a 1 a. With a additioal iitial coditio 0, the sequece becomes the Pell umber sequece: {p 0, 1,, 5, 1, 9,...}, which also satisfies the recurrece relatio p p 1 p.usig formula.3 i Corollary., we obtai the geeral term of the Pell umber sequece: p 1 { 1 ) 1 ) }..17 The umerators of the closest ratioal approximatio to are half the compaio Pell umbers or Pell-Lucas umbers. By addig i iitial coditio, we obtai the Pell-Lucas umber sequece {c,, 6, 14, 34, 8,...}, which satisfies c c 1 c. Similarly, Corollary. gives c 1 ) 1. ).18 We ow cosider the sequece of the sums of Pell umber: {σ satisfies the recurrece relatio σ σ 1 σ 1. 0, 1, 3, 8, 0,...}, which.19 From Remark.3, the above expressio ca be trasfered to a equivalet form b b 1 b,.0 where b σ 1/. Usig Corollary., oe easily obtai b 4{ 1 1 ) 1 1 ) 1 }..1

6 6 Iteratioal Joural of Mathematics ad Mathematical Scieces Thus, σ 4{ 1 1 ) 1 1 ) 1 } 1.. If the coefficiets of the liear recurrece relatio of a fuctio sequece {a x } of order are real or complex-value fuctios of variable x, thatis, a x p x a 1 x q x a x,.3 we obtai a fuctio sequece of order with iitial coditios a 0 x ad a 1 x. I particular, if all of p x, q x, a 0 x, ad a 1 x are polyomials, the the correspodig sequece {a x } is a polyomial sequece of order. Deote the solutios of t p x t q x 0.4 by α x ad β x. The α x 1 ) p x p x 4q x, β x 1 ) p x p x 4q x..5 Similar to Propositio.1, we have Propositio.7. Let {a } be a sequece of order satisfyig the liear recurrece relatio.3. The ) ) a1 x β x a 0 x α a1 x α x a 0 x x β x, α x β x α x β x a x a 1 x α 1 x 1 a 0 x α x, if α x / β x ; if α x β x,.6 where α x ad β x are show i.5. Example.8. Cosider the Chebyshev polyomials of the first kid, T x, defied by T x cos arc cos x,.7 which satisfies the recurrece relatio T x xt 1 x T x.8

7 Iteratioal Joural of Mathematics ad Mathematical Scieces 7 with T 0 x 1adT 1 x x. Thus the correspodig p, q, α, ad β are, respectively, x, 1, x x 1, ad x x 1, which yields T 1 x βt 0 x x 1, T 1 x αt 0 x x 1, ad α β x 1. Substitutig the quatities ito.7 yields T x 1 { ) ) } x x 1 x x 1..9 All the Chebyshev polyomials of the secod kid, third kid, ad fourth kid satisfy the same recurrece relatioship as the Chebyshev polyomials of the first kid with the same costat iitial term 1. However, they possess differet liear iitial terms, which are x, x 1, ad x 1, respectively see, e.g., Maso ad Hadscomb 10 ad Rivli 11. We will give the expressio of the Chebyshev polyomials of the secod kid later by sortig them ito the class of the geeralized Gegebauer-Humbert polyomials. As for the Chebyshev polyomials of the third kid, T 3 x, ad the Chebyshev polyomials of the fourth kid, x, whe x / 1, we clearly have the followig expressios usig a similar argumet T 4 preseted for the Chebyshev polyomials of the first kid: T 3 x T 4 x x 1 x 1 x 1 x 1 x 1 x 1 x x 1 x x 1 ) x 1 x 1 x 1 ) x 1 x 1 x 1, x x 1).30 ). x x 1 Example.9. I 1, Adré-Jeai studied the geeralized Fiboacci ad Lucas polyomials defied, respectively, by U x; a, b U x a U 1 bu, U 0 x 0, U 1 x 1, V x; a, b V x a V 1 bv, V 0 x, V 1 x x a,.31 where a ad b are real parameters. Clearly, V x;0, 1 T x. UsigPropositio.7, we obtai { 1 ) } U x; a, b x a x a x 4b) x a x a 4b, a 4b V x; a, b 1 ) } {x a x a 4b) x a x a 4b..3 From the last expressio, we also see V x;0, 1 T x. A sequece of the geeralized Gegebauer-Humbert polyomials {P λ,y,c x } 0 is defied by the expasio see, e.g., Gould 13 ad He et al. 14 : Φ t C xt yt λ P λ,y,c x t,.33 0

8 8 Iteratioal Joural of Mathematics ad Mathematical Scieces where λ>0, y ad C / 0 are real umbers. As special cases of.33, we cosider P λ,y,c x as follows see 14 : P 1,1,1 x U x, Chebyshev polyomial of the secod kid, P 1/,1,1 x ψ x, Legedre polyomial, P 1, 1,1 P 1, 1,1 x P 1 x, Pell polyomial, x ) F 1 x, Fiboacci polyomial, P 1,,1 x ) Φ 1 x, Fermat polyomial of the first kid,.34 P 1,a, x D x, a, Dickso polyomial, where a is a real parameter ad F F 1 is the Fiboacci umber. Theorem.10. Let x / ± Cy. The geeralized Gegebauer-Humbert polyomials {P 1,y,C x } 0 defied by expasio.33 ca be expressed as P 1,y,C x C x x Cy x Cy x x Cy ) x x Cy x Cy ) x x Cy..35 Proof. Takig derivative with respect to x to the two sides of.33 yields λ x yt ) C xt yt λ 1 P λ,y,c x t The, substitutig the expasio of C xt yt λ of.33 ito the left-had side of.36 ad comparig the coefficiets of term t o both sides, we obtai C 1 P λ,y,c 1 By trasferrig 1, we have x x λ P λ,y,c x y λ 1 P λ,y,c 1 x..37 P λ,y,c x x λ 1 C P λ,y,c 1 x y λ C P λ,y,c x.38 for all with P λ,y,c 0 x Φ 0 C λ, P λ,y,c 1 x Φ 0 λxc λ 1..39

9 Iteratioal Joural of Mathematics ad Mathematical Scieces 9 Thus, if λ 1, P 1,y,C x satisfies liear recurrece relatio P 1,y,C x x C P 1,y,C 1 x y C P 1,y,C x,, P 1,y,C 0 x C 1, P 1,y,C 1 x xc..40 Therefore, we solve t pt q 0, where p x/c ad q y/c, fort, ad obtai solutios: α 1 C β 1 C { } x x Cy, { } x x Cy,.41 where x / ± Cy. Hece, Propositio.7 gives the formula of P 1,y,C x as P 1,y,C x C x x Cy x Cy α x x Cy x Cy β,.4 where α ad β are show as.41. This completes the proof. Remark.11. We may use recurrece relatio.40 to defie various polyomials that were defied usig differet techiques. Comparig recurrece relatio.40 with the relatios of the geeralized Fiboacci ad Lucas polyomials show i Example.9, with the assumptio of P 1,y,C 0 0adP 1,y,C 1 1, we immediately kow that P 1,1,1 x xp 1,1,1 1,1,1 x P x U x;0, defies the Chebyshev polyomials of the secod kid, P 1, 1,1 x xp 1, 1,1 x P 1, 1,1 x U x;0, defies the Pell polyomials, ad P 1, 1,1 x ) xp 1, 1,1 x ) 1 P 1, 1,1 x ) U x;0, 1.45 defies the Fiboacci polyomials. I additio, i 15, Lidl et al. defied the Dickso polyomials are also the special case of the geeralized Gegebauer-Humbert polyomials, which ca be defied uiformly usig recurrece relatio.40, amely, D x; a xd 1 x; a ad x; a P 1,a, x.46

10 10 Iteratioal Joural of Mathematics ad Mathematical Scieces with D 0 x; a add 1 x; a x. Thus, the geeral terms of all of above polyomials ca be expressed usig.35. Example.1. For λ y C 1, usig.35, we obtai the expressio of the Chebyshev polyomials of the secod kid: U x x ) 1 x 1 x ) 1 x 1 x 1,.47 where x / 1. Thus, U x 4x 1. For λ C 1ady 1, formula.35 gives the expressio of a Pell polyomial of degree 1: P 1 x x ) 1 x 1 x ) 1 x 1 x Thus, P x x. Similarly, let λ C 1ady 1, the Fiboacci polyomials are F 1 x ad the Fiboacci umbers are x ) 1 x 4 x ) 1 x 4 1 x 4,.49 F F 1 1 { ) ) } 1 5,.50 which has bee preseted i Example.5. Fially, for λ C 1ady, we have Fermat polyomials of the first kid: Φ 1 x x ) 1 x x ) 1 x x,.51 where x /. From the expressios of Chebyshev polyomials of the secod kid, Pell polyomials, ad Fermat polyomials of the first kid, we may get a class of the geeralized Gegebauer-Humbert polyomials with respect to y defied as follows. Defiitio.13. The geeralized Gegebauer-Humbert polyomials with respect to y, deoted by P y x, are defied by the expasio 1 xt yt 1 P y x t,.5 0

11 Iteratioal Joural of Mathematics ad Mathematical Scieces 11 by P y x xp y x yp y x, 1.53 or equivaletly, by P y x ) 1 x x y x x y ) 1 x y.54 with P y 0 x 1adP y 1 x x, where x / y. I particular, P 1 x, P 1 x, ad P x are, respectively, Pell polyomials, Chebyshev polyomials of the secod kid, ad Fermat polyomials of the first kid. 3. Idetities Costructed from Recurrece Relatios From. we have the followig result. Propositio 3.1. A sequece {a } 0 of order satisfies liear recurrece relatio.1 if ad oly if it satisfies the ohomogeeous liear recurrece relatio of order 1 with the form a αa 1 dβ 1, 3.1 where d is uiquely determied. I particular, if β 1, thea αa 1 d is equivalet to a α 1 a 1 αa,whered a 1 αa 0. Proof. The ecessity is clearly from.1. We ow prove sufficiecy. If sequece {a } satisfies the ohomogeeous recurrece relatio of order 1 show i 3.1, the by substitutig 1 ito the above equatio, we obtai d a 1 αa 0.Thus, 3.1 ca be writte as a αa 1 a 1 αa 0 β 1, 3. which implies that {a } satisfies the liear recurrece relatio of order : a pa 1 qa with p α β ad q αβ. I particular, if β 1, the p α 1adq α, which yields the special case of the propositio. A obvious example of the special case of Propositio 3.1 is the Mersee umber a 1 0, which satisfies the liear recurrece relatio of order : a 3a 1 a with a 0 0ada 1 1 ad the ohomogeeous recurrece relatio of order 1: a a 1 1 with a 0 0. It is easy to check that sequece a k 1 / k 1 satisfies both the homogeeous recurrece relatio of order, a k 1 a 1 ka, ad the ohomogeeous recurrece relatio of order 1, a ka 1 1, where a 0 0ada 1 1.

12 1 Iteratioal Joural of Mathematics ad Mathematical Scieces We ow use 3. to prove some idetities of Fiboacci ad Lucas umbers ad geeralized Gegebauer-Humbert polyomials. Let {F } be the Fiboacci sequece. From 3., F αf 1 F 1 αf 0 β 1 β 1, 3.3 where the last step is due to α β 1. Therefore, we give a simple idetity F αf 1 β F 1 ) 1 1 5, 3.4 which is show i 16, 8. page 1 by Koshy. Similarly, we have βf βαf 1 β β F 1, 3.5 where the last step is due to αβ 1. The above idetity ca be writte as ) F F The same argumet yields α αf F 1, or equivaletly, ) F F Idetities 3.6 ad 3.7 were proved by usig differet method i 16, page 78. Let {L } be the Lucas umber sequece with L 0 adl 1 1, which satisfies recurrece relatio. with the same α ad β for the Fiboacci umber sequece. The, usig the same argumet, we have L αl 1 L 1 αl 0 β 1 1 α β 1 5β Thus ) 1 5 L L 1 ) , 3.9 or equivaletly, ) 1 5 L 1 L ) see 16, page 19.

13 Iteratioal Joural of Mathematics ad Mathematical Scieces 13 We ow exted the above results regardig Fiboacci ad Lucas umbers to more geeral sequeces preseted by Nive et al. i 17. Let {G } 0 ad {H } 0 be two sequeces defied, respectively, by the liear recurrece relatios of order : G pg 1 qg, H ph 1 qh 3.11 with iitial coditios G 0 0adG 1 1adH 0 adh 1 p, respectively. Clearly, if p q 1, the G ad H are, respectively, Fiboacci ad Lucas umbers. From 3., we immediately have G p p 4q p p 4q G Multiplyig p p 4q / to both sides of the above equatio yields p p 4q p p 4q G qg Similarly, we obtai p p 4q p p 4q G qg Whe p q 1, the last two idetities are 3.6 ad 3.7, respectively. Usig 3. we ca also obtai the idetity p p 4q H H 1 p p p 4q 4q, 3.15 which implies 3.10 whe p q 1. Aharoov et al. see 18 have proved that the solutio of ay sequece of umbers that satisfies a recurrece relatio of order with costat coefficiets ad iitial coditios a 0 0ada 1 1 ca be expressed i terms of Chebyshev polyomials. For istace, the authors show F i U i/ ad L i T i/. Thus, we have idetities ) i U ) ) i i T ) ) i i i 1 U, ) 1 5 i T 1 i 1 ). 3.16

14 14 Iteratioal Joural of Mathematics ad Mathematical Scieces I 19, Che ad Louck obtaied F 1 i U i/. Thus we have idetity ) 1 5 i ) i U 1 ) i i U Idetities 3.6 ad 3.7 ca be used to prove the followig radical idetity give by Sofo i 0 : αf F 1 1/ 1 1 F 1 αf 1/ Idetity 3.7 shows that the first term o the left-had side of 3.18 is simply α. Assume the sum i the third parethesis o the left-had side of 3.18 is c, the βc βf 1 αβf F βf 1 β 1, 3.19 where the last step is from 3.6 with trasform 1. Thus, we have c β.if is odd, the left-had side of 3.18 is α β 1. If is eve, the left-had side of 3.18 becomes α β α β 1, which completes the proof of Sofo s idetity. I geeral, let {a } be a sequece of order satisfyig liear recurrece relatio 1.1 or equivaletly.1. The we sum up our results as follows. Theorem 3.. Let {a } 0 be a sequece of umbers or polyomials defied by the liear recurrece relatio a pa 1 qa 0) with iitial coditios a 0 ad a 1, ad let p α β ad q αβ. The we have idetity a αa 1 a 1 αa 0 β I particularly, if {a P λ,y,c x }, the sequece of the geeralized Gegebauer-Humbert polyomial is defied by.33, the we obtai the polyomial idetity: P 1,y,C x αp 1,y,C 1 x C x αc β 1, 3.1 where α ad β are show i.41. For C 1 i.e., the geeralized Gegebauer-Humbert polyomials with respect to y), we deote P y x P 1,y,1 x ad have ) P y x x x y P y 1 x x x y). 3. Actually, 3. ca also be proved directly. Similarly, for the Chebyshev polyomials of the first kid T x, we have the idetity ) 1. x x 1 T 1 x T x x 1 x x 1) 3.3

15 Iteratioal Joural of Mathematics ad Mathematical Scieces 15 Let x cos θ, the above idetity becomes e iθ cos 1 θ cos θ i si θe i 1 θ, 3.4 which is equivalet to cos θ cos 1 θ cos θ si 1 θ si θ. Aother example is from the sequece {a } show i Corollary. with a 0 1ad a 1 p. The 3.0 gives the idetity a αa 1 β, where α p p 4 /adβ 1/α. Similar to 3.6 ad 3.7, for those sequeces {a } with a 0 1ada 1 p, we obtai idetities p p 4 1 p p 4 a a 1, p p 4 1 p p 4 a a Whe p 1, the above idetities become 3.6 ad 3.7, respectively. Similarly, we ca prove a k α k a β a k, where α p p 4 /adβ 1/α. It is clear that if 1/a is bouded ad β < 1, from 3.0 we have lim α. a 1 a 3.6 Therefore, lim F L lim F 1 L 1 The method preseted i this paper caot be exteded to the higher-order settig. However, we may use the idea ad a similar argumet to derive some idetities of sequeces of order greater tha. For istace, for a sequece {a } of umbers or polyomials that satisfies the liear recurrece relatio of order 3: a pa 1 qa ra 3, 3, 3.8 we set the equatio t 3 pt qt r Usig trasform t s p/3, we ca chage the equatio to the stadard form s 3 as b 0, which ca be solved by Vieta s substitutio s u a/ 3u. The formulas for the three roots, deoted by α, β, γ, are sometimes kow as Cardao s formula. Thus, we have α β γ p, αβ βγ γα q, αβγ r. 3.30

16 16 Iteratioal Joural of Mathematics ad Mathematical Scieces Deote y a αa 1. The 3.8 ca be writte as y β γ ) y 1 βγy From Propositios.1 or.7, oe may obtai ) ) y1 βy 0 y1 γ γy 0 β, if γ / β; γ β γ β y y 1 γ 1 1 y 0 γ, if γ β. 3.3 Therefore, from the idetity y γy 1 β 1 y 1 γy 0, we obtai idetity i terms of a : β 1 y 1 γy 0 ) a αa 1 γ a 1 αa a α γ ) a 1 αγa, 3.33 or equivaletly, a α γ ) a 1 αγa β 1 a 1 α γ ) a 0 αγa 1 ), 3.34 where a 1 ca be foud uiquely from y β γ y 1 βγy 0 ad y 0 a 0 αa 1,thatis, a αa 1 β γ ) a 1 αa 0 βγ a 0 αa 1, 3.35 or equivaletly, a 1 a pa 1 qa 0, r/ r We have see the equivalece betwee the homogeeous recurrece relatio of order 3, i 3.8, ad the ohomogeeous recurrece relatio of order, i Remark 3.3. Similar to the particular case show i Propositio 3.1, we may fid the equivalece betwee the ohomogeeous recurrece relatio of order, a pa 1 qa k, ad the homogeeous recurrece relatio of order 3, a p 1 a 1 q p a qa 3, where k a pa 1 qa 0.

17 Iteratioal Joural of Mathematics ad Mathematical Scieces 17 Example 3.4. As a example, we cosider the triboacci umber sequece geerated by a a 1 a a 3 3. Solvig t 3 t t 1 0, we obtai β / /3, α i ) /3 1 1 i ) /3, 6 6 γ i ) /3 1 1 i ) / Substitutig α, β, γ,ady 0 0 with the assumptio a 1 0 ad y 1 1ito 3.33,weobtai a idetity regardig the triboacci umber sequece {a 0, 1, 1,, 4, 7, 13, 4,...}: a 1 3 a b a a b a b ) a a b 1, 3.38 where a /3 ad b /3. For the sequece defied by a 6a 1 11a 6a 3, 3, 3.39 with iitial coditios a 0 a 1 1ada. The, the first few umbers of the sequece are {1, 1,, 7, 6, 91,...}. The three roots of t 3 6t 11t 6 0areα 1, β, ad γ 3. Therefore, by assumig a 1 7/6, we obtai the correspodig y 0 1/6ady 1 0adthe followig idetity for the above-defied sequece: a 4a 1 3a 3.40 { 1 } { } for all. From 1 by Haye, a 1 0, where are Stirlig umbers of 3 k the secod kid. Hece, we obtai a idetity of the Stirlig umbers of the secod kid: { } { } { } The idea to reduce a liear recurrece relatio of order 3 to order ca be exteded to the higher order cases. I geeral, if we have a sequece {a } satisfyig the liear recurrece relatio of order r: r a p k a k. k 1 3.4

18 18 Iteratioal Joural of Mathematics ad Mathematical Scieces Assume the equatio t r r k 1 p kt r k 0 has solutios α k 1 k r. Deote y a α 1 a 1. The the above recurrece relatio ca be reduced to r r y y 1 α k y α i α j y 3 α i α j α k 1 r y r 1 α k, k i / j r i / j / k r k 3.43 a liear recurrece relatio of order r 1 for sequece {y }. Usig this process, we may obtai the explicit formula of a ad/or idetities i terms of a if we kow the solutio of the last equatio ad/or the idetities i terms of sequece {y }. The process show i Propositio 3.1 ca be applied coversely to elevate a ohomogeous recurrece relatio of order to a homogeeous recurrece relatio of order Solutios of Algebraic Equatios ad Differetial Equatios The results preseted i Sectios ad 3 have more applicatios. I this sectio, we will discuss the applicatios i the solutios of algebraic equatios ad iitial value problems of secod-order ordiary differetial equatios. First, we cosider roots of polyomials p x x xf F 1 or the solutio of p x 0, where {F } 0 is the Fiboacci sequece. Usig the idetity α αf F 1, we immediately kow that the largest root of p x is x α Ideed, p x oly chages its coefficiet sigs oce, which implies that it has oly oe positive root α ad all of its other roots must be egative, for example, β 1 5 /. I, Wall proved the largest root of p x is α usig a more complicated maer. We may write the idetity α αf F 1 as ) ) F F 1 L 5F, 4. where L is the th Lucas umber. Similarly, we have ) ) F F 1 L 5F. 4.3 Multiplyig the last two equatios side by side yields 1 L 5F /4, or equivaletly, L F 4 1. The last expressio meas 5F 4 1 is a perfect square, or equivaletly, if is a Fiboacci umber, the 5 ± 4 is a perfect square. This result is a part of Gessel s results i 3, but the method we used seems simpler. I additio, the above result also shows that the Pell s equatio x 5y ±4 has a solutio x, y L,F. The above results o the Fiboacci sequece ca be exteded to the sequece {G } 0 show i 17 ad Sectio 3. Cosider polyomial p x x xg qg 1 with q 1. We ca

19 Iteratioal Joural of Mathematics ad Mathematical Scieces 19 see the largest root of p x is p p 4q /, which implies Wall s result whe p q 1. I additio, because of p p 4q p p 4q G qg 1 p p 4q 1 p 4qG H ), p p 4q G qg ) p 4qG H, we have q 1 4 ) ) H p 4q G, 4.5 or equivaletly, H ) p 4q G 4 q. 4.6 Hece, p 4q G 4 q is a perfect square, which implies the special case for the Fiboacci sequece that has bee preseted. Therefore, Pell s equatio x p 4q y ±4q p 4q >0 has a solutio x, y H,G. We ow use the method preseted i Sectio to reduce a iitial problem of a secod order ordiary differetial equatio: y py qy 0, y 0 A, y 0 B, 4.7 of secod-order ordiary differetial equatio with costat coefficiets to the problem of liear equatios. Let p, q / 0, ad let α ad β be solutio s of x px q 0. Deote v x : y x αy x. 4.8 The v x y x αy x, ad the origial iitial problem of the secod order is split ito two problems of first order by usig the method show i. for : v x βv x, v 0 A αb; y x αy x v x, y 0 B. 4.9

20 0 Iteratioal Joural of Mathematics ad Mathematical Scieces Thus, we obtai the solutios v x A αb e βx, 1 ) A βb e αx A αb e βx), if α / β; y x α β e αx A αb x B, if α β The above techique ca be exteded to the iitial problems of higher-order ordiary differetial equatios. I this paper, we preseted a elemetary method for costructio of the explicit formula of the sequece defied by the liear recurrece relatio of order ad the related idetities. Some other applicatios i solutios of algebraic ad differetial equatios ad some extesios to the higher dimesioal settig are also discussed. However, besides those applicatios, more applicatios i combiatorics ad the combiatorial explaatios of our give formulas still remai much to be ivestigated. Ackowledgmets Dedicated to Professor L. C. Hsu o occasio of his 90th birthday. The authors wish to thak the referees for their helpful commets ad suggestios. Refereces 1 T. Masour, A formula for the geeratig fuctios of powers of Horadam s sequece, The Australasia Joural of Combiatorics, vol. 30, pp. 07 1, 004. A. F. Horadam, Basic properties of a certai geeralized sequece of umbers, The Fiboacci Quarterly, vol. 3, pp , L. Comtet, Advaced Combiatorics: The Art of Fiite ad Ifiite Expasios, D. Reidel, Dordrecht, The Netherlads, L. C. Hsu, Computatioal Combiatorics, Shaghai Scietific & Techical, Shaghai, Chia, 1st editio, G. Strag, Liear Algebra ad Its Applicatios, Academic Press, New York, NY, USA, d editio, H. S. Wilf, Geeratigfuctioology, Academic Press, Bosto, Mass, USA, A. T. Bejami ad J. J. Qui, Proofs that Really Cout: The Art of Combiatorial Proof, vol. 7 of The Dolciai Mathematical Expositios, Mathematical Associatio of America, Washigto, DC, USA, C. Falbo, The golde ratio a cotrary viewpoit, The College Mathematics Joural,vol.36,o.,pp , G. E. Bergum, L. Beett, A. F. Horadam, ad S. D. Moore, Jacobsthal polyomials ad a cojecture cocerig Fiboacci-like matrices, The Fiboacci Quarterly, vol. 3, pp , J. C. Maso ad D. C. Hadscomb, Chebyshev Polyomials, Chapma & Hall/CRC, Boca Rato, Fla, USA, T. J. Rivli, Chebyshev Polyomials: From Approximatio Theory to Algebra ad Number Theory, Puread Applied Mathematics New York, Joh Wiley & Sos, New York, NY, USA, d editio, R. Adré-Jeai, Differetial properties of a geeral class of polyomials, The Fiboacci Quarterly, vol. 33, o. 5, pp , H. W. Gould, Iverse series relatios ad other expasios ivolvig Humbert polyomials, Duke Mathematical Joural, vol. 3, pp , T.-X. He, L. C. Hsu, ad P. J.-S. Shiue, A symbolic operator approach to several summatio formulas for power series. II, Discrete Mathematics, vol. 308, o. 16, pp , R. Lidl, G. L. Mulle, ad G. Turwald, Dickso Polyomials, vol. 65 of Pitma Moographs ad Surveys i Pure ad Applied Mathematics, Logma Scietific & Techical, Harlow, UK; Joh Wiley & Sos, New York, NY, USA, 1993.

21 Iteratioal Joural of Mathematics ad Mathematical Scieces 1 16 T. Koshy, Fiboacci ad Lucas Numbers with Applicatios, Pure ad Applied Mathematics New York, Wiley-Itersciece, New York, NY, USA, I. Nive, H. S. Zuckerma, ad H. L. Motgomery, A Itroductio to the Theory of Numbers, Joh Wiley & Sos, New York, NY, USA, 5th editio, D. Aharoov, A. Beardo, ad K. Driver, Fiboacci, Chebyshev, ad orthogoal polyomials, The America Mathematical Mothly, vol. 11, o. 7, pp , W. Y. C. Che ad J. D. Louck, The combiatorial power of the compaio matrix, Liear Algebra ad Its Applicatios, vol. 3, pp , A. Sofo, Geeralizatio of radical idetity, The Mathematical Gazette, vol. 83, pp , R. L. Haye, Biary relatios o the power set of a -elemet set, Joural of Iteger Sequeces, vol. 1, o., article 09..6, 009. C. R. Wall, Problem 3, The Fiboacci Quarterly, vol., o. 1, p. 7, I. Gessel, Problem H-187, The Fiboacci Quarterly, vol. 10, o. 4, pp , 197.