Matrix representations of Fibonacci-like sequences

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1 NTMSCI 6, No. 4, New Treds i Mathematical Scieces Matrix represetatios of Fiboacci-like sequeces Yasemi Tasyurdu Departmet of Mathematics, Faculty of Sciece ad Art, Uiversity of Erzica Biali Yıldırım, Erzica 4000, Turkey Received: 0 October 08, Accepted: 9 October 08 Published olie: 3 December 08. Abstract: I this paper, we defie Fiboacci-Like matrix sequeces of Fiboacci-Like sequeces associated with k-pell, k-pell Lucas ad Modified k-pell sequeces ad the we derive the matrices give th geeral term of these matrix sequeces. We also obtai Biet formulas, geeratig fuctios ad some fudametal idetities ivolvig terms of these matrix sequeces. Keywords: k-pell sequece, k-pell Lucas sequece, Modified k-pell sequeces, Matrix. Itroductio There are so may articles i the literature that cocer the special umber sequeces such as Fiboacci, Lucas, Pell, Pell-Lucas, Modified Pell, Jacobsthal, Jacobsthal-Lucas satisfyig secod order liear recurrece relatios ad it has may importat applicatios to diverse fields such as mathematic, computer sciece, physics, biology ad statistics [], [4] [5], [9], [], [3]. Also, it has bee obtaied a lot of studies o the ew family of k-fiboacci, k-pell, k-lucas, k-jacobsthal umbers [6], [7], [8], [4]. The sequeces W a,b; p,q are defied by the geeral recurrece relatio W pw qw, with W 0 a, W b ad a, b, p, q, itegers with p, q 0 by Horadam [6], [7], [8]. I fact, i the Horadam otatio, it ca be writte k-pell sequeces { P k,, k-pell Lucas sequeces { Qk, ad Modified k-pell sequeces { qk, respectively by P k, W 0,;,k Q k, W,;,k 3 The correspodig characteristic equatio of equatio is q k, W,;,k. 4 x pxq ad its roots are r p+ p 4q ad r p p 4q. For equatios, 3 ad 4, their roots are r + +k ad r +k ad verify r + r, r r +k ad r r k. Correspodig author ytasyurdu@erzica.edu.tr c 08 BISKA Bilisim Techology

2 04 Y. Tasyurdu: Matrix represetatios of Fiboacci-like sequeces Biet formulas of k-pell sequeces, k-pell Lucas sequeces ad Modified k-pell sequeces are give respectively by P k, r r r r Q k, r + r q k, r + r. For ay positive real umber k, Fiboacci-Like sequeces { R k, associated with k-pell sequeces, k-pell Lucas sequeces ad Modified k-pell sequeces are defied recurretly by R k, R k, + kr k,, 5 with R k,0 ad R k, [5]. Clearly x xk0 is also the characteristic equatio of R k, R k, + kr k, recurrece formula i equatio 5 ad a ad b are its two roots where a+ +k ad b +k. May properties of these umber sequeces are deduced directly from elemetary matrix algebra. The matrix method is used to get some properties for these umber sequeces. It has bee cosidered ew matrices which are based o Fiboacci ad Pell sequeces [], [], [3], [0], []. The mai aim of the preset article is to itroduce matrix represetatios of Fiboacci-Like sequeces which is similar to k-pell, k-pell-lucas, Modified k-pell sequeces ad kow as Fiboacci-Like matrix sequeces. Mai Results. Fiboacci-Like Matrix Sequeces I this sectio, we itroduce defiitio of the Fiboacci-Like matrix sequeces by usig Fiboacci-Like sequeces associated with k-pell, k-pell-lucas, Modified k-pell sequeces. Throughout this paper, the symbol R k, is th term of Fiboacci-Like matrix sequeces associated with k-pell, k-pell- Lucas, Modified k-pell sequeces ad ay positive real umber k. Defiitio.For ay positive real umber k, Fiboacci-Like matrix sequeces are defied recurretly by +k with iitial coditios R k,0, R k 3 k,. k k Example.A few terms of Fiboacci-Like matrix sequeces are R k, R k, + k R k,, 6 R k,0, k 3 +k R k, k k, 4+5k +k R k, k+k k, R k,3 8+k+k 4+5k 4k+5k k+k... c 08 BISKA Bilisim Techology

3 NTMSCI 6, No. 4, / 05 Theorem. For 0 ad ay positive real umber k, we have Rk,+ R R k, k, kr k, kr k, 7 where R k, is th term of Fiboacci-Like sequeces. Proof. We ca use the iductio method o. Let us cosider R k, 3 k, R k,0, R k,,r k, +k from equatio 5. For 0 ad, we have Rk, R R k,0 k,0 kr k,0 kr k, R k, Rk, R k, kr k, kr k,0 k 3 +k k k So the proof is completed for 0 ad. By iteratig this procedure ad cosiderig iductio steps, let us assume that the equatio 7 holds for all Z +. To fiish the proof, we have to show that the equatio 7 holds for +. By our assumptio,. Hece, we obtai the desired result. R k,+ R k, + k R k, Rk,+ R k, Rk, R + k k, kr k, kr k, kr k, kr k, Rk,+ + kr k, R k, + kr k, kr k, + k R k, kr k, + k R k, Rk,+ R k,+ kr k,+ kr k, The th geeral term of Fiboacci-Like matrix sequeces ca be foud by usig the followig theorem. Theorem. For 0 ad ay positive real umber k, the th term of Fiboacci-Like matrix sequeces is give by k, λ R k,0 R R k, R λ k, λ R k,0 λ λ λ λ λ. Proof. The characteristic equatio of R k, R k, + k R k, recurrece formula is λ λ k 0. The solutios of this equatio are λ ad λ. The geeral term of Fiboacci-Like matrix sequeces may be expressed i the form R k, Aλ + Bλ 8 for some coefficiets A ad B. Givig to the values 0, ad solvig this system of liear equatios, it is obtaied Usig A ad B i equatio 8, we obtai R k, λ R k,0 R k, λ R k,0 A ad B λ λ λ λ R k, R k, λ R k,0 λ λ λ R k, λ R k,0 λ λ λ c 08 BISKA Bilisim Techology

4 06 Y. Tasyurdu: Matrix represetatios of Fiboacci-like sequeces which is as desired. Also, the roots λ ad λ verifies the relatio such as where λ + +k ad λ +k. λ λ k λ + λ. Geeratig fuctios for Fiboacci-like matrix sequeces I this sectio, we give geeratig fuctios for Fiboacci-Like matrix sequeces. A geeratig fuctio gx is a formal power series gx i0 whose coefficiets give the sequece {a 0,a,a,... Give a geeratig fuctio is the aalytic expressio for the th term i the correspodig series. We will show that Fiboacci-Like matrix sequeces ca be cosidered as the coefficiets of the power series of the correspodig geeratig fuctio. Let us suppose that the terms of Fiboacci-Like matrix sequeces are coefficiet of a potetial series coutered at a x the origi ad cosider the correspodig aalytic fuctios r k x such that r k x R k,0 + R k, x+ R k, x R k, x The we ca write r k xx R k,0 x+ R k, x + R k, x R k, x From the equatios 9, 0 ad, we obtai where R k,i R k,i + k R k,i with iitial R k,0 k 3 fuctios of Fiboacci-Like matrix sequeces which is the desired. kr k xx k R k,0 x + k R k, x 3 + k R k, x k R k, x r k xr k xxkr k xx R k,0 + R k, x R k,0 x are +k, R k, from Defiitio. So the geeratig k k r k x R k,0 + R k, R k,0 x xkx.3 Some idetities of Fiboacci-like matrix sequeces I this sectio, we preset some of the iterestig properties of Fiboacci-Like matrix sequeces like Catala s idetity, Cassii s idetity, d Ocage s idetity. Theorem 3. For ay positive real umber k, c 08 BISKA Bilisim Techology R k,i i0 λ i λ R λ k, + λ λ R k,0 λ k λ R λ λ k,+ + k R k,. λ k

5 NTMSCI 6, No. 4, / 07 Proof. We ca use Theorem to prove. From Theorem, we obtai R k,i i0 λ i R k, λ R k,0 λ λ i0 i λ R k, λ R k,0 λ λ λ By usig the defiitio of a geometric sequece, λ + λ ad λ λ k, we have i0 i λ. λ R k,i R k, λ R k,0 i0 λ i λ + λ + R k, λ R k,0 λ + λ + λ λ λ + λλ λ λ λ λ + λλ λ R k, λ R k,0 λ + λ + λ λ λ λ λ R k, λ R k,0 λ + λ + λ λ λ λ λ R k, λ R k,0 λ + λ + λ λ λ λ λ λ λ λ λ R k, λ R k,0 λ + λ + λ λ λ λ λ λ λ λ λ R k, λ R k,0 λ + λ λ λ + λ λ λ k λ λ R k, λ R k,0 λ + λ λ λ + λ λ λ k λ λ If we rearrage the last equatio, the we get ad so we obtai which is the desired. ] R k,i [ R k, λ R k,0 R i0 λ i λ λ λ + k, λ R k,0 λ λ λ + λ λ λ k λ λ λ λ ] [ R k, λ R k,0 R λ λ λ + k, λ R k,0 λ λ + λ λ k λ λ λ λ ] [ R k, λ R k,0 R + λ λ λ + k, λ R k,0 λ λ + λ λ k λ λ λ λ λλ λ R k, +λ λ λ λλ + λ R k,0 λ λ k λ λ λ R λ λ k,+ λ λ R k,. λ k R k,i i0 λ i λ R λ k, + λ λ R k,0 λ k λ R λ λ k,+ + k R k, λ k c 08 BISKA Bilisim Techology

6 08 Y. Tasyurdu: Matrix represetatios of Fiboacci-like sequeces Theorem 4. For ay positive real umber k ad j > m, we get i0 R k,mi+ j R k,m+m+ j k m R k,m+ j +k m R k, jm R k, j λ m+ λ m. km Proof. Let us take U R k, λ R k,0 ad V R k, λ R k,0. The we get i0 R k,mi+ j i0 Uλ mi+ j Vλ mi+ j λ λ Uλ j λ λ λ mi Vλ j i0 Uλ j λ m+ λ λ λ m Uλ m+m+ j Vλ m+m+ j λ λ λ mi i0 Uλ m+ j Vλ j Vλ m+ j λ m+ λ m λ λ λ λ m + If we use Theorem, λ + λ ad λ λ k, we obtai which is as desired. i0 R k,mi+ j Uλ jm λ m + λ m λ λ m λ λ λ λ m Vλ jm R k,m+m+ j k m R k,m+ j +k m R k, jm R k, j λ m + λ m km Uλ j Vλ j λ λ Theorem 5. R k, R k,m R k,m R k,. Proof. We ca use Theorem to prove R k, R k,m R k,m R k,. From Theorem, we get R k, R k,m Rk,+ R k, kr k, kr k, Rk,m+ R k,m kr k,m kr k,m Rk,+ R k,m+ + kr k, R k,m R k,+ R k,m + kr k, R k,m kr k, R k,m+ + k R k, R k,m kr k, R k,m + k R k, R k,m R k,+ R k,m+ + kr k, R k,m Rk, + kr k, Rk,m + kr k, R k,m kr k, Rk,m + kr k,m + k R k, R k,m kr k, R k,m + k R k, R k,m R k,m+ R k,+ + kr k,m R k, Rk,m + kr k,m Rk, + kr k,m R k, kr k,m Rk, + kr k, + k R k, R k,m kr k, R k,m + k R k, R k,m Rk,m+ R k,+ + kr k,m R k, R k,m+ R k, + kr k,m R k, kr k,m R k,+ + k R k,m R k, kr k,m R k, + k R k,m R k, Rk,m+ R k,m Rk,+ R k, kr k,m kr k,m kr k, kr k, R k,m R k,. So the proof is completed. As the other way, Theorem ca be used for proof of Theorem 5 too. Theorem 6. Catala s Idetity For 0 r, R k,r R k,+r R k,. c 08 BISKA Bilisim Techology

7 NTMSCI 6, No. 4, / 09 where ay positive real umber k. Proof. Let us take U R k, λ R k,0 ad V R k, λ R k,0. By usig Theorem, we ca write R k,r R k,+r R k, Uλ r If we arrage the last equatio, we obtai Vλ r λ λ Uλ +r Vλ +r λ λ Uλ Vλ. λ λ R k,r R k,+r R k, UV λ λ λ r λ +r λ +r λ r λ λ. O the other had, we get UV R k, λ R k,0 R k, λ R k,0 [0] x +k by usig R k,0 ad R k 3 k, from Defiitio. Hece, we obtai k k which is the required result. R k,r R k,+r R k, Note that for r i obtaied Catala s idetity, we have the Cassii s idetity for Fiboacci-Like matrix sequeces. So we ca write followig corollary. Corollary. Cassii s Idetity R k, R k,+ R k,. Theorem 7. For ay real umber k ad <m, s t, R k,m+t R k,+s R k,m+s R k,+t. Proof. Let us take U R k, λ R k,0 ad V R k, λ R k,0. By usig Theorem, we obtai Uλ m+t Vλ R k,m+t R k,+s R m+t Uλ +s Vλ +s k,m+s R k,+t λ λ λ λ Uλ m+s Vλ m+s Uλ +t Vλ +t λ λ λ λ U λ m+t++s UVλ m+t λ +s UVλ +s λ m+t +V λ m+t++s λ λ U λ m+t++s +UVλ m+s λ +t +UVλ +t λ m+s V λ m+t++s + λ λ UVλ t λ s λ s λ t λ λ m λ m λ λ λ. Cosequetly, from UV R k, λ R k,0 R k, λ R k,0 [0] x, we get as required. R k,m+t R k,+s R k,m+s R k,+t Note that for t 0 ad s i Theorem 7, we obtai d Ocage s idetity. So we ca write followig corollary. c 08 BISKA Bilisim Techology

8 0 Y. Tasyurdu: Matrix represetatios of Fiboacci-like sequeces Corollary. d Ocage s Idetity For < m, R k,m R k,+ R k,m+ R k, [0] x. Ackowledgemet The author declare that they have o competig iterests. The author express their sicere thaks to the referee for his/her careful readig ad suggestios that helped to improve this paper. Refereces [] N. Bickell, A primer o the Pell sequece ad related sequece, Fiboacci Quarterly, 34, 975, [] A. Dasdemir, O the Pell, Pell-Lucas ad Modified Pell Numbers by Matrix Method, Applied Mathematical Scieces, 564, 0, [3] J. Ercolao, Matrix geerators of Pell sequeces, The Fiboacci Quarterly, 7, 979,7-77. [4] V. E. Hoggatt, Fiboacci ad Lucas Numbers. A publicatio of the Fiboacci Associatio, Bosto, Houghto Mi- i Compay, OCoLC , 969. [5] A. F. Horadam, Applicatios of Modified Pell Numbers to Represetatios, Ulam Quart., 3, 994, [6] A. F. Horadam, Basic properties of a certai geeralized sequeces of umbers, Fiboacci Quarterly, 33, 965, [7] A. F. Horadam, Geeratig fuctios for powers of umbers, Duke Math. J. 33, 965, [8] A. F. Horadam, Special properties of the sequece w a,b; p,q, Fiboacci Quarterly, 54, 967, [9] A. F. Horadam, Jacobsthal Represetatio Numbers, The Fiboacci Quarterly, 34, 996, [0] D. Kalma, Geeralized Fiboacci umbers by matrix methods, The Fiboacci Quarterly, 0, 98, [] E. Kılıç ad D. Taşçı, The Liear Algebra of The Pell Matrix, Bol. Soc. Mat. Mexicaa, 3; 005. [] T. Koshy, Fiboacci ad Lucas Numbera with Applicatios, Wiley- Itersciece Publicatios, 00. [3] N. N. Vorobiov, Números de Fiboacci, Editora MIR, URSS, 974. [4] Y. Tasyurdu, N. Cobaoglu ad Z. Dilme, O The a New Family of k-fiboacci Numbers, Erzica Uiversity Joural of Sciece ad Thechology, 9, 06, [5] A. A. Wai, S. A. Bhat ad G.P.S. Rathore, Fiboacci-Like Sequeces Associated With k-pell, k-pell-lucas ad Modified k-pell Sequeces, Joural of Applied Mathematics ad Computatioal Mechaics, 6, 07, c 08 BISKA Bilisim Techology