THE GENERALIZED (s, t)-fibonacci AND FIBONACCI MATRIX SEQUENCES

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1 TJMM 7 205), No 2, THE GENERALIZED s, t)-fibonacci AND FIBONACCI MATRIX SEQUENCES AHMET İPEK, KAMIL ARI, AND RAMAZAN TÜRKMEN Abstract In ths paper, we study the generalzatons of the s, t)-fbonacc and Lucas sequences and the s, t)-fbonacc and Lucas matrx sequences We present relatonshp between the s, t)-fbonacc matrx and generalzed Fbonacc matrx sequences Bnet s formula for the generalzed s, t)-fbonacc matrx sequence s derved We establsh several denttes for the generalzed s, t)-fbonacc and Fbonacc matrx sequence We gve some partal sum formulas for the generalzed s, t)-fbonacc and Fbonacc matrx sequence Also, we fnd out relatonshp between the s, t)-fbonacc matrx sequence and the famous Bernoull numbers Introducton The lterature ncludes many papers dealng wth the specal number sequences such as Fbonacc, Lucas, Pell, Jacobsthal, Mersenne, Fermat, Padovan, Perrn see, eg, -7] and the references cted theren) The books wrtten by Hoggat 6], Koshy 0] and Vajda 3] collects and classfes many results dealng wth the these number sequences, most of them are obtaned qute recently These numbers have been generalzed n many ways -5, 7, 9,, 4, 5] Falcón and Plaza 4] defned the followng one-parameter generalzaton of the Fbonacc sequence n whch t generalzes both the classc Fbonacc sequence and the Pell sequence, and deduced many propertes of these numbers from elementary matrx algebra Defnton 4]) For any nteger number k, the k-fbonacc sequence, say {F k,n } n N wth ntal condtons F k,n+ kf k,n + F k,n for n F k,0 0; F k, However, two-parameters generalzatons of the Fbonacc and Lucas sequences, respectvely, are gven n ] and 2] by Defnton 2 ]) For any nteger numbers s > 0 and t 0 wth s 2 + 4t > 0; the nth s, t)-fbonacc sequence, say {F n s, t)} n N wth F 0 s, t) 0, F s, t) F n+ s, t) sf n s, t) + tf n s, t) for n, ) Defnton 3 2]) For any nteger numbers s > 0 and t 0 wth s 2 + 4t > 0; the nth s, t)-lucas sequence, say {L n s, t)} n N L n+ s, t) sl n s, t) + tl n s, t) for n, 200 Mathematcs Subject Classfcaton B39, 40C05, B68, B50 Key words and phrases Fbonacc sequence, Fbonacc matrx sequence, Bnet s formula, partal sums, Bernoull numbers 37

2 38 AHMET İPEK, KAMIL ARI, AND RAMAZAN TÜRKMEN wth L 0 s, t) 2, L s, t) s The followng table summarzes specal cases of F n s, t) and L n s, t) : s, t) F n L n, ) Fbonacc numbers Lucas numbers 2, ) Pell numbers Pell-Lucas numbers, 2) Jacobsthal numbers Jacobsthal-Lucas numbers 3, 2) Mersenne numbers Fermat numbers On the other hand, the matrx sequences that concern ths specal number sequences have taken so much nterest, 2, 5, 6, ] In ] and 2], t s defned new matrx generalzatons of the Fbonacc and Lucas sequences such that: Defnton 4 ]) For any nteger numbers s > 0 and t 0 wth s 2 + 4t > 0; the nth s, t)-fbonacc matrx sequence, say {F n s, t)} n N wth F 0 s, t) F n+ s, t) sf n s, t) + tf n s, t) for n, 2) 0 s and F 0 s, t) t 0 Defnton 5 2]) For any nteger numbers s > 0 and t 0 wth s 2 + 4t > 0; the nth s, t)-lucas matrx sequence, say {L n s, t)} n N L n+ s, t) sl n s, t) + tl n s, t) for n, s 2 s wth L 0 s, t), L 2t s s, t) 2 + 2t s st 2t They showed some propertes of these matrx sequences usng essentally a matrx approach n ] and 2] Moreover, n that papers, varous denttes based on the matrx representatons has been derved for the nth s, t)-fbonacc and Lucas sequences Our goal n ths paper s to generalze several results about the s, t)-fbonacc and Lucas matrx sequences and s, t)-number sequences such as Fbonacc, Lucas, Pell, Jacobsthal, Mersenne and Fermat numbers The purpose of ths paper s to obtan some properts of the generalzed s, t)-fbonacc matrx sequence and to demonstrate that some propertes of the generalzed s, t)-fbonacc numbers and the matrces defned n ] and 2] are vald for a more general sequence of matrces, ntroduced n Secton 2 In the rest of the paper: -): the matrx sequence n whch t generalzes the s, t)-fbonacc and Lucas matrx sequences wll be defned -): the relatonshp between these matrx sequences wll be presented -): by gvng the the Bnet formulas and summaton formulas over these new matrx sequence, some fundamental propertes on the generalzed s, t)-fbonacc numbers wll be obtaned v-): usng Bnet s formula for the generalzed s, t)-fbonacc matrx sequence, the relatonshp between the s, t)-generalzed Fbonacc matrx sequence and the famous Bernoull numbers wll be nvestgated Therefore, by the generalzed s, t)-fbonacc matrx sequence defned n Secton 2, we have a great opportunty to obtan some new propertes over the s, t)-generalzed Fbonacc numbers defned n Secton 2

3 FIBONACCI MATRIX SEQUENCES 39 2 The generalzed s, t)-fbonacc matrx sequence In ths secton, we wll manly focus on the s, t)-generalzed Fbonacc sequence and the generalzed s, t)-fbonacc matrx sequence to get some mportant results Defnton 6 For any nteger numbers s > 0 and t 0 wth s 2 + 4t > 0; the nth s, t)-generalzed Fbonacc sequence, say {G n s, t)} n N G n+ s, t) sg n s, t) + tg n s, t) for n, 3) where G 0 s, t) a 0 and G s, t) a, wth a 0, a R In ths paper, we wll denote smply G n, F n and L n nstead of G n s, t), F n s, t) and L n s, t), respectvely Defnton 7 For any nteger numbers s > 0 and t 0 wth s 2 + 4t > 0; the nth generalzed s, t)-fbonacc matrx sequence, say {R n s, t)} n N wth R 0 s, t) R n+ s, t) sr n s, t) + tr n s, t), n, 4) a a 0 sa + ta and R ta 0 a sa s, t) 0 a 0 ta ta 0 Agan, n ths paper we denote R n nstead of R n s, t) From the recurrence relatons n ) and 2), Cvcv and Turkmen obtaned Fn+ F F n n ] 5) tf n tf n Smlarly, from the recurrence relatons n 3) and 4), we obtan, for n 0, Gn+ G R n n 6) tg n tg n Theorem ]) F m+n F m F n for any ntegers m, n 0 Lemma For n 0 holds: R n+ R F n 7) Proof We use the second prncple of fnte nducton on n to prove ths lemma When n 0, snce F 0 I, the result s true Let n Then the lemma yelds R 2 R F, whch defnes the matrx R 2 Now assume that R n+ R F n for n N Then R F N+ R F N F by Theorem R N+ F by our assumpton Gn+2 G n+ s tg n+ tg n t 0 R N+2 Thus t s true for every nonnegatve nteger n

4 40 AHMET İPEK, KAMIL ARI, AND RAMAZAN TÜRKMEN 2 Bnet formula for the generalzed s, t)-fbonacc matrx sequence Bnet s formulas are well known n the Fbonacc and Lucas numbers theory 0] Bnet s formulas for the recurrence relaton ) s wdely used for smulaton of varous physcal and bologcal phenomena In our case, Bnet s formula allows us to express the s, t)-generalzed Fbonacc matrx sequences n functon of the roots α and β of the characterstc equaton x 2 sx + t, assocated to the recurrence relaton ), or 3) Theorem 2 ]) F n F βf 0 α n F αf 0 β n, n 0, 8) Corollary ]) The nth s, t)-fbonacc number s gven by F n αn β n 9) Corollary 2 The nth generalzed s, t)-fbonacc matrx sequence s gven by R2 βr R n+ α n R2 αr β n, n 0 0) Proof From the formula 8) and Lemma, the proof s completed Corollary 3 The nth s, t)-generalzed Fbonacc number s gven by G n a a 0 β αn a a 0 α βn ) Proof The proof of ths corollary s tral from Corollary 2 22 Identtes for the generalzed s, t)-fbonacc matrx sequence usng Bnet s formula In ths secton, we derve several denttes for the generalzed s, t)-fbonacc and Fbonacc matrx sequences by smple matrx algebra Corollary 4 R m+n+ R m+ F n for any ntegers m, n 0 Proof The proof s obvous from Theorem and Lemma Corollary 5 G m+n+2 G m+2 F n+ + tg m+ F n Proof From 5), 6) and Corollary 4, we have Gm+n+2 G m+n+ Gm+2 G m+ Fn+ F n tg m+n+ tg m+n tg m+ tg m tf n tf n From where, we obtan the result For s t, we obtan the convoluton formula of Honsberger Theorem 3 F n r F n+r F 2 n 2)

5 FIBONACCI MATRIX SEQUENCES 4 Proof Let X F βf 0 and Y F αf 0 By usng Eq 8) and takng nto account that αβ t and Y X XY t s obtaned F n r F n+r Fn 2 Xαn r Y β n r Xα n+r Y β n+r Xα n Y β n ) 2 X2 α 2n XY α n r β n+r XY α n+r β n r + Y 2 β 2n X 2 α 2n + 2XY α n β n Y 2 β 2n ) 2 r r ] ) 2 αβ) n β XY αβ) n α XY + 2 αβ) n XY α β αβ)n+ α 2r + β 2r ] ) 2 αβ) r 2 XY t) n+ r F 2 nxy by Eq 9) Thus, snce XY 0, Eq 2) s proven Corollary 6 F 2n+ F n r+ F n+r+ + tf n r F n+r, 3) F 2n F n r+ F n+r + tf n r F n+r 4) Proof From Eq 5) and Theorem, we have Fn r+ F F n r F n+r n r Fn+r+ F n+r tf n r tf n r tf n+r tf n+r and F 2 n F2n+ F 2n tf 2n tf 2n From Theorem 3, snce the terms a and a 2 of both sdes of F n r F n+r Fn 2 are equal, the Eq 3) and 4) are obtaned Corollary 7 R n r+ F n+r R n+ F n Proof The proof s tral from Lemma and Theorem 3 Corollary 8 G n r+2 F n+r+ + tg n r+ F n+r G n+2 F n+ + tg n+ F n, 5) G n r+2 F n+r + tg n r+ F n+r G n+2 F n + tg n+ F n 6) Proof From Eq 5) and 6), we get Gn r+2 R n r+ F n+r G n r+ ) Fn+r+ F n+r and R n+ F n tg n r+ tg n r tf n+r tf n+r Gn+2 G n+ Fn+ F n tg n+ tg n tf n tf n ) From Corollary 7, snce the terms a and a 2 of both sdes of R n r+ F n+r R n+ F n are equal, the Eq

6 42 AHMET İPEK, KAMIL ARI, AND RAMAZAN TÜRKMEN It s known that the lmt of the rato of two adjacent Fbonacc numbers as well as the adjacent Lucas numbers and the adjacent numbers of any numercal sequence that s gven by the recurrence relaton )) tends to the postve characterstc root, e, f n lm + 5, n f n 2 where f n s the nth classcal Fbonacc number The Golden properton s the postve root of the characterstc equaton x 2 x +, that s also called the Golden secton equaton The Hadamard product, or the Schur product, of two matrces A and B of the same sze s defned to be the entrywse product A B a j b j ) The Hadamard unt matrx U s such a matrx whose all entres are the sze of U beng understood ) 8] The matrx A called Hadamard nvertble f all ts entres are non-zero Then A a j s called as the Hadamard nverse of A 2] An useful property n the s, t)-generalzed Fbonacc matrx sequences s that the lmt of the Hadamard quotent of two consecutve terms s equal to αu, where U s the Hadamard unt matrx Theorem 4 lm R n+ R n αu 2, 7) n where U 2 s the 2 2 Hadamard unt matrx Proof Let X R 2 βr and Y R 2 αr By usng Eq 0), lm R n+ R n lm n n Xαn Y β n ) Xα n Y β n ) n ) β lm X Y n α α X n ) β Y, β α n and takng nto account that lm β n α) 0 snce β < α, Eq 7) s obtaned Corollary 9 For the generalzed s, t)-fbonacc sequence, lm postve root of the characterst equaton x 2 sx + t G n n G n ) α where α s the 23 Partal Sums for the generalzed s, t)-fbonacc matrx sequence usng Bnet s formula In ths secton, we present partal sum formulas for the generalzed s, t)-fbonacc and Fbonacc matrx sequences usng Bnet s formula gven for F n Theorem 5 Let s + t and S n be the sum of the frst n + ) terms of the s, t)- Fbonacc matrx sequence, that s S n n F Then, S n 0 s + t F n+ + tf n F + s ) F 0 ] 8) Proof Let s + t and X F βf 0 and Y F αf 0 Consderng Eq 8), S n may be wrtten as S n n Xα Y β ) Now, by summng up to geometrc partal sums n α, or n β, we obtan 0 S n α n+ 0 0 α X βn+ β Y ) 9)

7 FIBONACCI MATRIX SEQUENCES 43 From where, after some algebra, we get S n α n α n+ β + ) X + β n + α + β n+ ) Y ] ) α ) β ) Xα n+ Y β n+ + t Xαn Y β n X Y ] αy + βx + s + t F n+ + tf n F 0 + α F ] αf 0 ) + β F βf 0 ) by 8) s + t s + t F n+ + tf n F + s ) F 0 ] wth α + β s, whch completed the poof Corollary 0 Let s + t Then, n R + s + t R n+2 + tr n+ R 2 + s ) R ] 20) 0 Proof The proof s tral from Lemma and Theorem 5 Corollary Let s + t Then, n G s + t ) G n+ + tg n a + s ) a 0 ] 0 Proof From Eq 20), by obtanng the term a 22 of the matrx s + t R n+2 + tr n+ R 2 + s ) R ] snce ths term s at the same tme n G from Eq 6), what we wanted s obtaned 0 By summng up the frst n + ) even terms of the s, t)-fbonacc matrx sequence we obtan: Theorem 6 Let s t + ) s + t ) 0 and T n be the sum of the frst n + ) even terms of the s, t)-fbonacc matrx sequence, that s T n n F 2 Then, T n s t + ) s + t ) 0 F2n+2 t 2 F 2n sf + s 2 + t ) F 0 ] Proof The proof s smlar to the proof of Theorem 5, and we only show an outlne of t Let s t + ) s + t ) 0, X F βf 0 and Y F αf 0 and T n n F 2 Replacng α α 2 and β β 2 n Eq 9), we have T n ) α 2 n+ ) β 2 n+ α 2 X β 2 Y 2) Snce, from 2) we obtan T n α 2 ) β 2 ) t ) 2 s 2, s t + ) s + t ) F 2n+2 t 2 F 2n F 0 + β2 X α 2 ] Y 0

8 44 AHMET İPEK, KAMIL ARI, AND RAMAZAN TÜRKMEN From where, we get snce T n Thus, the proof s completed F2n+2 t 2 F 2n sf + s 2 + t ) ] F 0, s t + ) s + t ) β 2 X α 2 Y sf + s 2 + t ) F 0 Corollary 2 Let s t + ) s + t ) 0 Then, n R 2+ R2n+3 t 2 R 2n+ sr 2 + s 2 + t ) ] R 22) s t + ) s + t ) 0 Proof The proof of corollary s obvous from Lemma and 6 Corollary 3 Let s t + ) s + t ) 0 Then, n G 2 G2n+2 t 2 G 2n sa + s 2 + t ) ] a 0 s t + ) s + t ) 0 Proof From Eq 22), by obtanng the term a 22 of the matrx R2n+3 + R 2n+ sr 2 + s 2 + t ) ] R s t + ) s + t ) snce ths term s at the same tme n G 2 from Eq 6), what we wanted s obtaned 0 Now, consderng Theorem 5 and 6 t s rghtly obtaned the sum of the frst odd terms of the s, t)-fbonacc matrx sequence: Theorem 7 Let s t + ) s + t ) 0 Then, n F 2+ F2n+3 t 2 ] F 2n+ + t ) F stf 0 s t + ) s + t ) 0 Corollary 4 Let s t + ) s + t ) 0 Then, n R 2+2 R2n+4 t 2 ] R 2n+2 + t ) R 2 str s t + ) s + t ) 0 Corollary 5 Let s t + ) s + t ) 0 Then, n G 2+ G2n+3 t 2 ] G 2n+ + t ) a sta 0 s t + ) s + t ) 0 In a smlar way, many formulas for partal sums of the term of the s, t)-fbonacc matrx sequence may be obtaned and partcularzed for dfferent values of s and t Theorem 8 n 0 n s t n F F 2n

9 FIBONACCI MATRIX SEQUENCES 45 Proof Let X F βf 0 and Y F αf 0 By Eq 8), we have n n s t n F n n Xs t n α Y s t n β ) 0 Thus, the result s obtaned Corollary 6 Corollary 7 0 X n 0 n t n sα) Y X t + αs)n Y t + βs) n ) n 0 ] n )t n sβ) Xα2n Y β 2n snce α 2 αs + t and β 2 βs + t F 2n n 0 n s t n R + R 2n+ 23) n 0 n s t n G G 2n Proof From Eq 23), by obtanng the term a 22 of the matrx R 2n+ snce ths term s at the same tme n ) s t n G from Eq 6), we get the result Theorem n n 0 n ) s n F F n+ F 0 F n F 24) Proof Let X F βf 0 and Y F αf 0 Then, we have, n n n ) n s n F ) Xα s n Y β ) whch completes the proof Corollary 8 n 0 0 X n 0 n s n α) Y X s α)n Y s β)n Xβn Y α n F 0 α n+ β n+) F α n β n ) F n+ F 0 F n F, by 8) n 0 n s n β) n ) s n R + F n+ R F n R 2 25)

10 46 AHMET İPEK, KAMIL ARI, AND RAMAZAN TÜRKMEN Proof The proof s obvous from Lemma and Theorem 9 Corollary 9 n 0 n ) s n G a 0 F n+ a F n Proof From Eq 25), by obtanng the term a 22 of the matrx F n+ R F n R 2 snce ths term s at the same tme n ) ) s n G, we have the result 0 n 24 More on dentty for the s, t)-fbonacc matrx sequence usng Bnet s formula In ths secton, we use Bnet s formula to study the relatonshp between the s, t)-generalzed Fbonacc matrx sequence and the famous Bernoull numbers Now, let X F βf 0 and Y F αf 0 From Eq 8) we easly deduce that the exponental generatng functon g x) s That s, g x) g x) eαx X e βx Y Therefore, we get or n0 n! F nx n n0 α n X β n Y ) n! x n 26) ) e βx e x s 2 +4t) X Y, snce s 2 + 4t s2 + 4t ) e βx e x s 2 +4t) X Y ) g x) e x s 2 +4t) x x s2 + 4t ), e x s 2 +4t) e βx Y e x s 2 +4t) X e kx s 2 +4t) g x) x x s2 + 4t ) 27) e x s 2 +4t) k0 On the other hand, the famous Bernoull numbers are defned by x e x x n B n, x < 2π 28) n! n0 A recurson formula nvolvng the Bernoull numbers are n n B n B k, k l0 k0 for n 2 and B 0, B 2 Then, from Eq 26) and 28) we have ) e βx Y e x s 2 +4t) ) X e lx s 2 +4t) F m B n m! xm x n s n! 2 + 4t) 29) From theory of nfnte seres t s well known that for two absolutely convergent power seres a n t n and b n t n, we wrte n0 n0 n0 n0 m0 n0 u+vn n0 ) ) a n t n b n t n a u b v t n,

11 FIBONACCI MATRIX SEQUENCES 47 so k tmes on both sdes of formula 29), we obtan LHS e βx Y e x s 2 +4t) X and RHS l0 e kβx Y e x s 2 +4t) X ) k ] k e lx s 2 +4t) l0 ) k e lx s 2 +4t) kβ) n ) k k Y e x s 2 +4t) X e lx s +4t)) 2 x n, n! n0 n0 u +u 2++u k +v +v 2++v k n) l0 l0 F u u! F u k B v ) v+v u k! v! B v 2++v k k s2 + 4 x n k v k! Comparng the coeffcents of x n k on the above, we mmedately obtan the followng dentty kβ) n k ) k ) k Y e x s 2 +4t) X e lx s 2 +4t) n k)! u +u 2++u k +v +v 2++v k n B v v! B v k v k! 3 Conclusons F u ) v+v u! F u 2++v k k s2 + 4 u k! In ths note, a new matrx generalzaton of the Fbonacc sequence, whch we call the generalzed s, t)-fbonacc matrx sequence, have been ntroduced and studed Usng the generalzed s, t)-fbonacc matrx sequence, many mathematcal formulas, whch allows us to express n a compact form the generalzed s, t)-fbonacc, have been gven References ] Cvcv, H and Turkmen, R, On the s, t)-fbonacc and Fbonacc matrx sequences, Ars Combnatora, ), ] Cvcv, H and Turkmen, R, Notes on the s, t)-lucas and Lucas matrx sequences, Ars Combnatora, ), ] Cureg, E and Mukherjea, A, Numercal results on some generalzed random Fbonacc sequences, Computers & Mathematcs wth Applcatons, 59 ) 200), ] Falcón, S and Plaza, A, On the Fbonacc k-numbers, Chaos, Soltons & Fractals, 32 5)2007) ] Gulec, HH and Taskara, N, On the s, t)-pell and s, t)- Pell-Lucas sequences and ther matrx representatons, Appled Mathematcs Letters, 25 0) 202), ] Hoggat, VE, Fbonacc and Lucas numbers, Palo Alto, Calforna, Houghton-Mffln, 969 7] Horadam, AF, A generalzed Fbonacc sequence, Math Mag 68 96), ] Horn, RA and Johnson, CR, Topcs n Matrx Analyss, Cambrdge Unversty Press, Cambrdge, 99 9] Klc, E, The Bnet formula, sums and representatons of generalzed Fbonacc p-numbers, European Journal of Combnatorcs, 29 3) 2008), ] Koshy, T, Fbonacc and Lucas Numbers wth Applcatons, John Wley & Sons, New York, NY, USA, 200 ] Ocal, AA, Tuglu, N, Altnsk, E, On the representaton of k-generalzed Fbonacc and Lucas numbers, Appled Mathematcs and Computatons, 70 ) 2005), ] Reams, R, Hadamard nverses, square roots and products of almost semdefnte matrces, Lnear Algebra and ts Applcatons, ), 35 43

12 48 AHMET İPEK, KAMIL ARI, AND RAMAZAN TÜRKMEN 3] Vajda, S, Fbonacc & Lucas numbers, and the golden secton Theory and Applcatons, Ells Horwood Lmted, 989 4] Yang, S, On the k-generalzed Fbonacc numbers and hgh-order lnear recurrence relatons, Appled Mathematcs and Computatons, 96 2) 2008), ] Yazlk, Y, Taskara, N, Uslu, K and Ylmaz, N, The Generalzed s, t)-sequence and ts Matrx Sequence, AIP Conference Proceedngs, ), ] Ylmaz, N, Taskara, N, Matrx Sequences n terms of Padovan and Perrn Numbers, Journal of Appled Mathematcs, Artcle ID ), 7 pages 7] Ylmaz, N, Taskara, N, Bnomal Transforms of the Padovan and Perrn Matrx Sequences, Abstract and Appled Analyss, Artcle ID ), 7 pages Karamanoğlu Mehmetbey Unversty, Faculty of Kaml Özdağ Scence Department of Mathematcs 7000 Karaman, Turkey E-mal address: ahmetpek@kmuedutr Karamanoğlu Mehmetbey Unversty, Faculty of Kaml Özdağ Scence Department of Mathematcs 7000 Karaman, Turkey E-mal address: kamlar@kmuedutr Selcuk Unversty, Scence Faculty Department of Mathematcs 42075, Campus, Konya, Turkey E-mal address: rturkmen@selcukedutr