Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 (ISSN: 2347-2529 Journal homepage: www.jaamm.com IJAAMM Internatonal Journal of Advances n Appled Mathematcs and Mechancs The bnomal transforms of the generalzed (s, t -Jacobsthal matr sequence Sukran UYGUN Mathematcs Department, Scence and Art Faculty, Gazantep Unversty, Turkey esearch Artcle eceved 11 November 2018; accepted (n revsed verson 11 January 2019 Abstract: In ths paper, we study the bnomal transforms of the generalzed (s, tjacobsthal matr sequence n+1 (s, t n N ; (s, t Jacobsthal J n+1 (s, t n N and, (s, t Jacobsthal Lucas C n+1 (s, t n N matr sequences. After thatby usng recurrence relatons of them, the generatng functons have beenfounded for these transforms. Fnally the relatons among these transforms have been demonstrated wth dervng new equaltes. MSC: 15A24 11B39 15B36 Keywords: Generalzed (s, t sequence Generalzed (s, t matr sequence (s, t Jacobsthal matr sequence (s, t Jacobsthal Lucas matr sequence. 2019 The Author(s. Ths s an open access artcle under the CC BY-NC-ND lcense (https://creatvecommons.org/lcenses/by-nc-nd/3.0/. 1. Introducton and prelmnary There are so many studes n the lterature that are concern about specal nteger sequences such as Fbonacc, Lucas, Pell, Jacobsthal, Padovan. You can encounter the generalzatons of these sequences n all of the references. In [1 the author wrote a book about these nteger sequences You can see the generalzed number and matr sequences for Fbonacc and Lucas sequences n [3, 5, 6. Smlarly the author defned number and matr sequences whch generalzes Jacobsthal and Jacobsthal Lucas sequences n [7, 8. Some authors ntroduced matr based transforms for these specal sequences. Bnomal transform s one of most popular transforms. You can have detaled nformaton about bnomal transform n [9, 10. Falcon defned dfferent bnomal transforms of thek Fbonacc sequence such as fallng, rsng bnomal transforms n [4. The authors gave bnomal transform for generalzed (s, t matr sequences n [11.The authors ntroduced bnomal transforms for the Padovan and Perrn numbers n [12. And n [13 bnomal transforms of the k Jacobsthal sequence a rentroduced. In [14 the authors gave some propertes of Lucas numbers wth bnomal coeffcents.the goal of ths paper s to apply the bnomal transforms to the generalzed Jacobsthal and Jacobsthal Lucas matr sequences. Also, the generatng functon of ths transform s found by recurrence relatons. Fnally the relatons among these transforms have been demonstrated wth dervng new equaltes. In [2, the author defned the Jacobsthal and Jacobsthal Lucas sequence as follows respectvely j n+1 = j n + 2j n 1 n 1, (j 0 = 0, j 1 = 1 c n+1 = c n + 2c n 1 n 1, (c 0 = 0, c 1 = 1 Now we gve some prelmnares related to our study. For a gven nteger sequence X = { 1, 2,..., n,...} the bnomal transform Y of the sequence X;Y (X = { } y n s defned by y n = =0 E-mal address: suygun@gantep.edu.tr. 14
Sukran UYGUN / Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 15 Defnton 1.1. Assume that a,b, s > 0, t 0 s 2 + 8t > 0. The(s, t Jacobsthal sequence {ĵ n (s, t} n N s defned by the followng recurrence relaton: ĵ n+1 (s, t = s ĵ n (s, t + 2t ĵ n 1 (s, t, (ĵ 0 (s, t = 0, ĵ 1 (s, t = 1 and, the (s, t Jacobsthal Lucas {ĉ n (s, t} n N s defned by the followng recurrence relaton ĉ n+1 (s, t = sĉ n (s, t + 2tĉ n 1 (s, t, (ĉ 0 (s, t = 2, ĉ 1 (s, t = s And the generalzed (s, tjacobsthal sequence {G n (s, t} n N s defned by the followng recurrence relaton G n+1 (s, t = sg n (s, t + 2tG n 1 (s, t, (G 0 (s, t = a, ĉ 1 (s, t = bs n [7. By choosng sutable values on a; b, we wll obtan (s, t Jacobsthal sequence;the (s, t Jacobsthal Lucas sequence by the generalzed (s, t Jacobsthal sequence: a = b = 1 { G n (s, t = ĵ n+1 (s, t a = 2, b = 1 {G n (s, t = ĉ n (s, t Defnton 1.2. Assume that a,b, s > 0, t 0 s 2 + 8t > 0. Generalzed (s, tjacobsthal matr sequence n+1 (s, t n N ; (s, t Jacobsthal J n+1 (s, t n N and, (s, t Jacobsthal Lucas C n+1 (s, t n N matr sequences are defned by the followng recurrence relatons n [8 : n+1 (s, t = s n (s, t + 2t n 1 (s, t (1 J n+1 (s, t = s J n (s, t + 2t J n 1 (s, t (2 C n+1 (s, t = sc n (s, t + 2tC n 1 (s, t (3 wth ntal condtons [ [ bs 2a bs 2 + 2at 2bs 0 = and at (b as 1 = bst 2at [ [ 1 0 s 2 J 0 (s, t = and J 0 1 1 (s, t = t 0 [ [ s 4 s 2 + 4t 2s C 0 (s, t = and C 2t s 1 (s, t = st 4t By choosng sutable values on a; b, we wll obtan (s, t Jacobsthal matr sequence;the (s, t Jacobsthal Lucas matr sequence by the generalzed (s, tjacobsthal matr sequence: a = b = 1 { n (s, t = J n+1 (s, t a = 2, b = 1 { n (s, t = C n (s, t In the rest of ths paper, for convenence we wll use the symbols ĵ n, ĉ n,g n, J n, C n, n nstead of ĵ n (s, t,ĉ n (s, t, G n (s, t, J n (s, t, C n (s, t, n (s, t respectvely. Proposton 1.1. The relatons between the number sequences and ther matr sequences are gven as [ [ [ ĵn+1 2ĵ n ĉn+1 2ĉ n Gn+1 2G n J n =,C n = and t ĵ n 2t ĵ n 1 tĉ n 2tĉ n =. n 1 tg n 2tG n 1
16 The bnomal transforms of the generalzed (s, t -Jacobsthal matr sequence 2. Bnomal transform of the (s, t Generalzed and (s, t Jacobsthal, (s, t Jacobsthal Lucas matr sequences In ths secton, the bnomal transforms of the generalzed (s, t matr sequence, (s, t Jacobsthal and (s, t Jacobsthal Lucas matr sequences wll be ntroduced. Defnton 2.1. Let n, J n and C n be the (s, t generalzed,(s, t Jacobsthal and (s, t Jacobsthal Lucas matr sequences, respectvely. The bnomal transforms of these matr sequences can be epressed as follows: The bnomal transform of the generalzed (s, t matr sequence s B n = =0 The bnomal transforms of (s, t Jacobsthal matr sequence s Ĵ n = J =0 The bnomal transforms of (s, t Jacobsthal Lucas matr sequence s Ĉ n = C =0 Lemma 2.1. For n 0, the followng equaltes are hold: B n+1 = n n ( + +1 =0 Ĵ n+1 = n =0 Ĉ n+1 = n =0 =1 (J + J +1 (C +C +1 Proof. Frstly, n here we wll just prove (, snce ( and ( can be proved ( by usng ( the same ( method.by ( usng the n + 1 n n n defnton of bnomal transform and the well-known bnomal equalty = + and = 0 t 1 n + 1 s obtaned that B n+1 = n+1 n + 1 =0 = 0 + n+1 [ n n + =1 1 = 0 + n+1 n + n n +1 = n =0 ( + +1 =0 whch s the desred result. Note that B n+1 s also wrtten as B n+1 = B n + n =0 +1 Theorem 2.1. For n 0, the sequences {B n }, { Ĵ n }, {Ĉn } are verfed the followng recurrence relatons a B n+2 = (s + 2B n+1 (s + 1 2tB n wth ntal condtons [ bs 2a B 0 = at (b as [ bs 2 + bs + 2at 2bs + 2a, B 1 = bst + at (b as + 2at
Sukran UYGUN / Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 17 b Ĵ n+2 = (s + 2Ĵ n+1 (s + 1 tĵ n [ [ 1 0 s + 1 2 Ĵ 0 = andĵ 0 1 1 = t 1 c Ĉ n+2 = (s + 2Ĉ n+1 (s + 1 tĉ n [ [ s 4 s 2 + 4t 2s Ĉ 0 = andĉ 2t s 1 = st 4t Proof. have We only prove the frst case because the other cases can be proved wth the same way. From Lemma 2.1, we n =0 = 0 + 1 + n B n+1 = n ( + +1 n ( =1 + +1 = 0 + 1 + (s + 1 n =1 =1 + 2t n =1 1 ± (s + 1 0 From Defnton 2.1, t s obtaned that B n+1 = (s + 1B n + 2t 1 + 1 s 0 (4 puttng n 1 nstead of n n (4 we have B n = (s + 1B n 1 + 2t n 1 n 1 =1 1 + 1 s 0 = sb n 1 + n 1 n 1 =0 + 2t n 1 n 1 =1 1 + 1 s 0 = sb n 1 + n [ n 1 n 1 + 2t =1 1 1 + 1 s 0 = sb n 1 + n [ n 1 n 1 n 1 + 2t ± 2t =1 1 1 1 + 1 s 0 = sb n 1 + n [ n 1 n (1 2t + 2t =1 1 1 + 1 s 0 = (s + 1 2tB n 1 + 2t n n 1 + 1 s 0 =1 =1 From ths equalty we have 2t 1 + 1 s 0 = B n (s + 1 2tB n 1 By substtutng ths epresson n (4, we obtan B n+1 = (s + 2B n (s + 1 2tB n 1 (5 whch completes the proof. The characterstc equaton of the bnomal transforms of the generalzed(s, t matr sequence B n s λ 2 (s+2λ+ (s 2t + 1 = 0.The roots of ths equaton are λ 1 = s + 2 + s 2 + 8t, λ 2 = s + 2 s 2 + 8t 2 2 Bnet formula are well known n the specal nteger sequences theory. Bnetformula allows us to epress the nth term n functon of the roots of λ 1 andλ 2 of the characterstc equaton, assocated the recurrence relaton (5. So the Bnet formula for B n can be epressed as B n = X λ 1 n Y λ 2 n λ 1 λ 2 [ bs 2 λ 2 bs + 2at 2bs 2aλ 2 X = bst λ 2 at 2at λ 2 (b as (6 [ bs 2 λ 1 bs + 2at 2bs 2aλ 1 Y = bst λ 1 at 2at λ 1 (b as (7 By choosng correspondng values on a and b n (6 and (7, we can obtan the Bnet formula of Ĵ n and Ĉ n. Namely,
18 The bnomal transforms of the generalzed (s, t -Jacobsthal matr sequence a For a = b = 1; we get the Bnet formula for the bnomal transforms of (s, t Jacobsthal matr sequence as Ĵ n+1 = Aλ 1 n n Bλ 2 where λ 1 λ 2 [ s 2 [ λ 2 s + 2t 2(s λ 2 bs 2 λ 1 s + 2t 2(s λ 1 A =, B = t(s λ 2 2t t(s λ 1 2t b For a =2, b = 1; we get the Bnet formula for the bnomal transforms of (s, t Jacobsthal Lucas matr sequence as Ĉ n = Cλ 1 n n Dλ 2 where λ 1 λ 2 [ s 2 [ λ 2 s + 4t 2(s 2λ 2 bs 2 λ 1 s + 4t 2(s 2λ 1 C =, D =. t(s 2λ 2 4t sλ 2 t(s 2λ 1 4t sλ 1 Theorem 2.2. The generatng functons of the bnomal transforms of generalzed (s, t matr sequence, (s, t Jacobsthal matr sequence, (s, t Jacobsthal Lucas matr sequence are B n (s, t, = = n=0 [ 1 1 (s+2+(s+1 t 2 J n (s, t, = = n=0 C n (s, t, = = B n n = 0+[ 1 (s+1 0 1 (s+2+(s+1 t 2 bs + (2at bs 2(a + (bs as a t(a + (bs as a (b as + (2at s 2 (b a s(b a ĵ n+1 n = J 0+[J 1 (s+1j 0 1 (s+2+(s+1 t 2 [ 1 s + (2t s 2(1 1 (s+2+(s+1 t 2 t(1 2t n=0 ĉ n n = C 0+[C 1 (s+1c 0 1 (s+2+(s+1 t 2 [ 1 s + (4t s 2(2 (s + 2 1 (s+2+(s+1 t 2 t(2 (s + 2 s + (4t + s 2 s Proof. We just prove the case ( and the others wll be omtted. Let B n (s, t, be generatng functon for the bnomal transform of generalzed (s, t Jacobsthal matr sequence. Then, B n (s, t, = B 0 + B 1 +... + n B n +... (8 If we multply (s + 2and (s + t 1 2 ; wth the both sdes of the equalty (8 respectvely, we obtan (s + 2 B n (s, t, = (s + 2 B 0 + (s + 2 2 B 1 +... + (s + 2 n+1 B n +... (9 (s + 1 2t 2 B n (s, t, = (s + 1 2t 2 B 0 + (s + 1 2t 3 B 1 +... + (s + 1 2t n+2 B n +... (10 Consderng (8, (9, (10 we get the followng equalty B n (s, t, ( 1 (s + 2 + (s + 1 t 2 = B 0 + (B 1 (s + 2B 0 (11 Fnally, from Theorem 2.1, Defnton 2.1, and (11 we have the desred result. We can get the followng relatons between the generalzed (s; t- matr sequence, (s, t Jacobsthal and (s, t Jacobsthal Lucas matr sequences and the generatng functons of the bnomal transforms of these sequences, respectvely. Let r ( = 0+[ 1 s 0 be the ordnary generatng functon of the sequence { 1 s t 2 n } : By usng the transformaton of 1 f 1 we have the generatng functon of the bnomal transform sequence {Bn } n Theorem 2.2-(. Let j ( = J 0+[J 1 s J 0 be the ordnary generatng functon of the sequence {J 1 s t 2 n }: By usng the transformaton of 1 j 1 we have the generatng functon of the bnomal transform sequence {Jn }n Theorem 2.2-(.
Sukran UYGUN / Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 19 Let c ( = C 0+[C 1 sc 0 be the ordnary generatng functon of the sequence {C 1 s t 2 n }: By usng the transformaton of 1 c 1 we have the generatng functon of the bnomal transform sequence{cn } n Theorem 2.2-(. Theorem 2.3. Let m, n N; then Ĵ m+n = Ĵ m Ĵ n. Proof. We use the nducton method. Let n = 0, then we get Ĵ m+0 = Ĵ m Ĵ 0 = Ĵ m I.Assume that Ĵ m+n = Ĵ m Ĵ n for n N. Then we obtan Ĵ m+n+1 = (s + 2Ĵ m+n (s + 1 tĵ m+n 1 = (s + 2Ĵ m Ĵ N (s + 1 tĵ m Ĵ N 1 = Ĵ m [ (s + 2 ĴN (s + 1 tĵ N 1 = Ĵ m Ĵ N+1 Theorem 2.4. Let n N, then n+1 = 1 J n J n+1 = J 1 J n C n+1 = C 1 J n Proof. The proof s easly obtaned by usng mathematcal nducton method. Theorem 2.5. The relatons among the transforms B n, Ĵ n and Ĉ n can be demonstrated by the followng equaltes: B n+1 B n = 1 Ĵ n Ĵ n+1 Ĵ n = J 1 Ĵ n Ĉ n+1 Ĉ n = C 1 Ĵ n Proof. By consderng Defnton 2.1, Lemma 2.1, we get B n+1 = n =0 By Theorem 2.4, B n+1 B n = n ( + +1 = B n + n =0 n +1 = =0 =0 +1. ( 1 J n = 1 Ĵ n Ths completes the proof of : The others are made by usng the same method. eferences [1 T. Koshy, Fbonacc and Lucas Numbers wth Applcatons, John Wleyand Sons Inc., NY (2001. [2 A. F. Horadam, Jacobsthal representaton numbers, The Fbonacc Quarterly.,34(1, (1996, 40-54. [3 S. Falcon and A. Plaza, The k-fbonacc sequence and the Pascal 2-trangle,Chaos, Soltons Fractals,33(2007, 38-49. [4 S. Falcon and A. Plaza, Bnomal Transforms of the k-fbonacc sequence,internatonal Journal of Nonlnear Scences and Numercal Smulaton,10(11-12 (2009, 1527-1538. [5 H. Cvcv,. Turkmen, On the (s, t Fbonacc and Fbonacc matr sequences, AS Combnatora, 87 (2008 161-173. [6 H. Cvcv,. Turkmen, Notes on the(s, t- Lucas and Lucas matr sequences, AS Combnatora, 89 (2008 271-285. [7 S. Uygun, The (s, t-jacobsthal and (s, t-jacobsthal Lucas sequences, Appled Mathematcal Scences, 70(9, (2015, 3467-3476.
20 The bnomal transforms of the generalzed (s, t -Jacobsthal matr sequence [8 K. Uslu, ÂÿS. Uygun, The (s, t-jacobsthal and (s, t-jacobsthal-lucas Matr sequences, AS Combnatora, 108, (2013 13-22. [9 H. Prodnger, Some nformaton about the bnomal transform, The Fbonacc Quarterly, 32(5, 1994, 412-415. [10 K. W. Chen, Identtes from the bnomal transform, Journal of NumberTheory, 124, 2007, 142-150. [11 Y. YazlÄśk, N. YÄślmaz, N. Taskara, The Generalzed (s, t-matr Sequences Bnomal Transforms, Gen. Math. Notes, 24(1,(2014,127-136. [12 N. YÄślmaz, N. Taskara, Bnomal transforms of the Padovan and Perrnnumbers, Journal of Abstract and Appled Mathematcs, (2013 ArtcleID941673. [13 S. Uygun, A. ErdoÄ du, Bnomnal transforms of k-jacobsthal sequences,journal of Mathematcal and Computatonal Scence, 7(6, (2017, 1100-1114. [14 N. Taskara, K. Uslu, H.H. Gulec, On the propertes of Lucas numbers wth bnomal coeffcents, Appled Mathematcs Letters, 23(1, (2010 68-72. Submt your manuscrpt to IJAAMM and beneft from: gorous peer revew Immedate publcaton on acceptance Open access: Artcles freely avalable onlne Hgh vsblty wthn the feld etanng the copyrght to your artcle Submt your net manuscrpt at edtor.jaamm@gmal.com