RELATIONS ON BI-PERIODIC JACOBSTHAL SEQUENCE

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1 TJMM , No., RELATIONS ON BI-PERIODIC JACOBSTHAL SEQUENCE S. UYGUN, H. KARATAS, E. AKINCI Abstrct. Following the new generliztion of the Jcobsthl sequence defined by Uygun nd Owusu 10 s ĵ n ĵ n 1 +ĵ n, when n is even nd ĵ n bĵ n 1 +ĵ n, when n is odd, with initil conditions ĵ 0 0, ĵ 1 for ll vlues of n. In this pper, through much broder study on this new generliztion, prticulrly considering its Binet formul, some numerous new identities nd properties of this sequence re investigted. 1. Introduction There re so mny rticles bout the specil integer sequences in the literure specilly bout the Fiboncci sequences nd generliztions of this sequence 1,, 3. But in the recent yers mny studies hve been seen bout the other sequences such s Jcobsthl sequence. The clssicl Jcobsthl sequence is defined s j n j n 1 + j n with initil condition j 0 0, j 1 1. Mny uthors studied bout generliztion of Jcobsthl numbers in 4. Bi-periodic Fiboncci sequence ws first introduced in literture on 009 by Edson nd Yyenie in 5. They gve the generting function, Cssini, Ctln nd D Ocgne properties for the bi periodic Fiboncci sequence etc. And then Yyenie found interesting properties of this sequence in 6. And lso Jun nd Choi in 11 gve some properties of this sequence by defining mtrix relted to bi periodic Fiboncci sequence. Just lie the bi-periodic Fiboncci sequence, Bilgici 7 introduced into literture the bi-periodic Lucs sequence nd gve some properties of bi-periodic Lucs sequence nd reltionship between bi-periodic Fiboncci sequence. In 8 the uthors defined bi-periodic Fiboncci mtrix sequence nd found nth generl term nd Binet formul of this mtrix sequence. Similrly in 9 the uthors investigted properties of bi-periodic Lucs mtrix sequence. In 10, the uthors defined bi-periodic Jcobsthl sequences similr to the bi-periodic Fiboncci nd Lucs sequences nd then proceed to find the bsic properties such s Binet formul, generting function, D Ocgne. In 11, the uthors defined bi-periodic Jcobsthl mtrix sequences. In this pper there will be much broder study on the properties of bi-periodic Jcobsthl sequences. The bi-periodic Jcobsthl sequence {ĵ n } n0 is defined s { ĵn 1 + ĵ ĵ 0 0, ĵ 1 1, ĵ n n, if n is even bĵ n 1 + ĵ n, if n is odd n, 1 where is the floor function of nd ξn n n is the prity function in 10, 11. Similrly we cn define the recurrence reltion by ĵ n+ 1 ξn b ξn ĵ n+1 + ĵ n. 010 Mthemtics Subject Clssifiction. 11B39, 11B83. Key words nd phrses. Generlized Jcobsthl sequence; Binet formul; Generting Functions. 141

2 14 S. UYGUN, H. KARATAS, E. AKINCI From the bove definition we hve the nonliner qudrtic eqution for the bi-periodic Jcobsthl sequence by x bx b 0. with roots α nd β defined by α b + b + 8b, β b b + 8b. 3 Some importnt properties of bi-periodic Jcobsthl sequence re given in the following proposition. By using the results we hve gret chnce to get different properties of bi-periodic Jcobsthl sequence. Proposition 1. The bi-periodic Jcobsthl sequence nd the roots re stisfied the following reltions The extended Binet formul The generting function ĵ m 1 ξm b m α m β m α β ĵx x1 + x x 1 b + 4x + 4x 4. 4 ĵ n+4 b + 4 ĵ n+ 4ĵ n 5 α + β α + β b, αβ b β + β b, α + α b α + β α, β + α β.. New Identities of Bi-periodic Jcobsthl Sequence Theorem 1. For ny nonnegtive integer n, it is obtined tht α n n ξn 1 b n ĵn α + n b n +ξnĵ n 1 β n n ξn 1 b n ĵn β + n b n +ξnĵ n 1 Proof. The identity holds for n 1. Let n ny integer. Using the extended Binet s formul, we hve ĵ n β 1 ξn α n β n b ĵn b n α β β 1 ξn α n β n b b n α β 1 ξn α n β n b n α β β α n β n α β 1 ξn α n α β b n α β 1 ξn b n 1 αn.

3 ĵ m+1 ĵ m 4ĵ m ĵ m 1 RELATIONS ON BI-PERIODIC JACOBSTHAL SEQUENCE 143 Since α bα b 0, multiplying by α b α + nd using αβ b, we hve ĵ n β b ĵn α + α b. 1 ξn b n 1 αn b n 1 ξn ĵ nα + ĵ n 4ĵ n α n α n n ξn 1 b n ĵn α + n b n +ξnĵ n 1. Theorem. For ny nonnegtive integer n, we hve ĵ n+6 b ξn b ξn ĵ n+3 + 8ĵ n Proof. The theorem cn be proved by using of the Binet s formul or the recurrence reltion s follows. For ny positive integer n, by nd 5 we hve Hence ĵ n+6 b + 4 ĵ n+4 4ĵ n+. ĵ n+6 b ξn b ξn ĵ n+3 + ĵ n+ 4ĵ n+ b ξn b ξn ĵ n+3 + b + 4 ĵ n+ b ξn b ξn ĵ n+3 + b + 4 ĵ n+ 1 ξn b ξn ĵ n+3 b ξn b ξn ĵ n+3 + b + 4 ĵ n+ 1 ξn b ξn 1 ξn b ξn ĵ n+ + ĵ n+1 b ξn b ξn ĵ n+3 + 8ĵ n When b 1 the bove result reduces to nown identity of Jcobsthl numbers ĵ n+6 7ĵ n+3 + 8ĵ n Theorem 3. For ny positive integer m > 1, we hve ĵ m 1 ĵ m+1 ĵ m + ĵ m 1 ĵ m Proof. Using Theorem 3 we get ĵ m+n 1 ξmn+n m 1 b 1 ξmn+n m ĵ m ĵ n + ξmn b ξmn ĵ m 1 ĵ n 1 ĵ m 1 ξm+1 b 1 ξmn+n m ĵ m+1 ĵ m 1 + α 1 ξm+1m 1 b ξm+1m 1 ĵ m ĵ m α 1 ξm b ξm ĵ m+1 ĵ m 1 + α 1 ξm+1 b ξm+1 ĵ m ĵ m ĵ m+1 α 1 ξm b ξm ĵ m 1 + α 1 ξm+1 b ξm+1 ĵ m ĵ m ĵ m+1 ĵ m ĵ m + α 1 ξm+1 b ξm+1 ĵ m ĵ m ĵ m+1 ĵ m ĵ m j m+1 α 1 ξm+1 b ξm+1 ĵ m

4 144 S. UYGUN, H. KARATAS, E. AKINCI Theorem 4. For ny positive integer m, it is obtined tht ĵ m ξm 1 0 m 1 b Proof. We will use the principle of induction to show the vlidity of the this formul stisfying for defining the bi-periodic Jcobsthl numbers by using sum formul. It is esily seen tht the ssertion is true for m 1, ĵ m 1 b Consider tht it is true for ny n such tht 1 n m. Then by the ssumption nd the property of binomil coefficients, it is obtined tht ĵ m+1 ξm b 1 ξm ĵ m + ĵ m 1 ξm b 1 ξm ξm 1 m + ξm 0 b ξm+1 0 m + ξm 0 0 m m 1 m ξm b ξm+1 m + ξm 0 ξm + m m m ξm m 1 0 m 1 m 1 b m b m b m +1 m 1 b b b m +1 b +ξm 1 b m 1 +1 b m m 1 b m

5 + m 1 RELATIONS ON BI-PERIODIC JACOBSTHAL SEQUENCE 145 m 1 1 ξm 1 0 b m m m ξm m b m + b m + m 1 ξm b m Theorem 5. Let m nd n be ny two positive integers, the following property is stisfied 1 ξmn+n m ξmn b b ĵ m+n 1 ĵ m ĵ n + ĵ m 1ĵ n 1. 7 Proof. We prove the bove result using the extended Binet s formul. First, note tht ξm + n ξm + ξn ξmξn. The first prt of the sum of the right hnd side is denoted by b 1 ξmn+n m 1 ξn b n α n β n α β 1 ξm b n ξm+n ξm+n 1+ 1 b α m+n +β m+n α n β m α m β n. b m+n α β α m β m And the second prt of the sum of the right hnd side is denoted by bξmn ξmn+1 ξn 1+1 ξm 1 α n 1 β n 1 α m 1 β m 1 b n + n α β α β α β ξm+n ξm+n 1+ 1 b α m+n + β m+n α n 1 β m 1 α m 1 β n 1 b m+n 1 α β. Now we dd the right hnd side of prt 7 ξm+n ξm+n 1+ 1 b b m+n ξm+n ξm+n 1+ 1 b α m+n 1 α + αbα 1 + β m+n 1 β + bβ 1 α β b m+n α β αm+n 1 α β β m+n 1 α β ξm+n ξm+n 1+ 1 b b m+n α m+n 1 β m+n 1 ĵ m+n 1 α β When b 1 the bove result reduces to n identity of Jcobsthl numbers ĵ m+n 1 ĵ m ĵ n + ĵ m 1 ĵ n 1. If we choose m n the theorem turns out the identity ĵ m 1 ĵ m + ĵ m 1. If we choose m m, n n + 1, the theorem turns out the identity ĵ m+n ĵ m ĵ n+1 + ĵ m 1 ĵ n.

6 146 S. UYGUN, H. KARATAS, E. AKINCI Theorem 6. For ny positive integer m >, we hve Proof. Since we get ĵ m ξm+1 m 1 m 0 m + 1 b m b + 8 α b + b b + 8, β b b b + 8 α m b + m m m b b β m b m m m b b + 8 Therefore, we obtin α m β m m m 0 m α m β m b b + 8 α m β m α β m b m b b m b b m b m j + 1 m j + 1 By using the definition of bi-periodic Jcobsthl sequence 1 ξm α m β m ĵ m b m α β 1 ξm b m ξm+1 m 1 ξm+1 m 1 j0 j0 1 m 1 j0 m j + 1 m j + 1 m j + 1 b m j 1 b + 8 j b m j 1 b + 8 j b m j 1 b + 8 j b m m j 1 b + 8 j b j b + 8 j When b 1 the bove result reduces to n identity of Jcobsthl numbers s ĵ m 1 m 1 j0 Theorem 7. For ny positive integer n, we hve n i1 ĵ i ĵ i+1 i 1 b m j + 1 ĵn+1 n 9 j 1 8

7 RELATIONS ON BI-PERIODIC JACOBSTHAL SEQUENCE 147 Proof. since therefore n i1 ĵ i ĵ i+1 i 1 ξi α i β i α i+1 β i+1 b i α β α β α i+1 + β i+1 αβ i α + β α β b i i α i β α β α + b b ĵ i ĵ i+1 i α β α α β α β α β α β α β 1 ξi+1 b i+1 i i β bi b b i n α i n β i n + β b 1 i b b i1 i1 i1 n 1 α i+1 n 1 β i+1 α + β b b i0 i0 α3 α 4n β 1 4n b n b α b 1 + β3 1 b n b β b 1 α α 4n b n b b n + β β 4n b n b b n α 4n+ + β 4n+ bb n b n+1 bb + 8 b α 4n+ + β 4n+ b n b n+1 bb + 8 α b α + αβ α α + β αb, β b β + αβ β α + β βb, α + β b + 4b b b + 4, α α α 4n b n α β b n + β β 4n b n b b n b α α 4n+ β 4n+ b + 4 α β 4 n n b b b + 8 The right hnd side of prt 8 is given by 1 ĵn+1 b n ξn+1 α n+1 β n+1 b b 1 n+1 α β n 4n+ + β 4n+ αβ n+1 b b n α β 1 4 n b

8 148 S. UYGUN, H. KARATAS, E. AKINCI 4n+ + β 4n+ b n+1 α β 4 n+1 b n+1 n b n+1 α β 4 1 n b 4n+ + β 4n+ b n+1 α β n b b b 4n+ + β 4n+ b n+1 α β 4 1 n b b + 4 b + 8 Theorem 8. For ny positive integer n, we obtin ξn+1 ξn b ĵ n ĵ n+ b 1 ξn+1 α n+1 β n+1 n+1. 9 n+1 b α β Proof. 1 ξn α n β n α n+ β n+ ĵ n ĵ n+ b n +1 α β ξn+1 αn+ + βn+ n αβ α + β b n ξn+1 α β ξn+1 α n+ + β n+ b n b + 4b b ξn+1 b n α β For the right hnd side of the equlity 9 b ξn+1 b αn+1 β n+1 ξn 1 ξn+1 n+1 n+1 b α β b ξn+1 b ξn αn+ + β n+ n+1 αβ ξn b n+1 ξn+1 α β n+1 1 α n+ + β n+ αβ n+1 + n+1 α β b n b ξn+1 b n α β 1 α n+ + β n+ b n+1 b n b + 8b b ξn+1 b n α β 1 α n+ + β n+ b n 4b + b + 8b b ξn+1 b n α β Theorem 9. For ny positive integer n, we hve n ξi ĵi ĵ i+ b i 1 ĵn+1 ĵ n+ b n Proof. b i1 ξi ĵi ĵ i+ ξi 1 ξi α i β i α i+ β i+ i b b +1 i α β i

9 RELATIONS ON BI-PERIODIC JACOBSTHAL SEQUENCE 149 ξi ξi+1 αi+ + βi+ i αβ α + β b b i ξi+1 α β i α i+ + β i+ b i b + 4b b α β b i similry n i1 ξi ĵi ĵ i+ b i b α β b α β b α β b α β b α β 1 ĵn+1 ĵ n+ b n 1 b α β b α β b α β n α i+ + β i+ b i b + 4b i1 b i { n α α b i + β β b i b + 4b } n 1 i i1 i1 { n 1 α α b i+1 + β β b i+1 b + 4b } n 1 i i0 i1 { α 3 α 4n αβ n b b n + β3 β 4n αβ n } b b n { α 4n+3 + β 4n+3 } bb n α 3 + β 3 1 b n 1 α n+1 β n+1 b n α β 1 α n+ β n+ b n+1 α β α 4n+3 + β 4n+3 αβ n+1 α + β b α β b n b 1 α 4n+3 + β 4n+3 b n α + βα αβ + β b n b α 4n+3 + β 4n+3 b b n α 3 + β 3 Theorem 10. For ny positive integer n, we hve n i1 ĵ i ĵ i+3 i+1 1 ĵn+1 ĵ n+3 b n+1 b Proof. Note tht ĵ i ĵ i+3 1 ξi α i β i 1 ξi+3 α i+3 β i+3 1 i+1 b i α β b i+3 α β i+1 α i+3 + β i+3 αβ i α 3 + β 3 b i + i+1 +1 α β i+1

10 150 S. UYGUN, H. KARATAS, E. AKINCI α β α α α β b i+1 + β β b i+1 i b b + 6 b i+1 α α b i+1 + β β b b + 6 b i+1 1 i n i1 ĵ i ĵ i+3 i+1 α β α β α β α β 1 α β α n i1 α b i+1 + β α b n 1 α b 1 n i1 α5 b + β β b i+1 b n 1 β b 1 b b + 6 β5 b α 4n b n α 4 b n+1 b + β4n b n β 4 b n+1 b α 4n b n α 4 b n+1 b + β4n b n β 4 b n+1 b α 4n+4 + β 4n+4 b b b n+1 b b b + 8 n i1 1 i It s written for the right hnd side of the eqution 10 1 ĵn+1 ĵ n+3 b n+1 b ξn+1 1 ξn+3 α n+1 β n+1 α n+3 β n+3 b b n+1 b n+3 α β b + 1 n+1 1 α 4n+4 + β 4n+4 αβ n+1 α + β b b n+1 α β b + 1 n+1 α 4n+4 + β 4n+4 b n+1 b + 4b 1 b b n+1 α β b α 4n+4 + β 4n+4 b α β b n+1 + bn+1 b b + 4 b b n+1 b b + 8 b + 1 b 1 α 4n+4 + β 4n+4 b + 4 b + 9b + 8 b α β n+1 + b b α 4n+4 + β 4n+4 b b b α β b n+1 b b + 8 References 1 Koshy, T., Fiboncci nd Lucs Numbers with Applictions, John Wiley nd Sons Inc., NY 001. George, A.H., Some formule for the Fiboncci sequence with generliztion, Fiboncci Qurt. 7, , 1969.

11 RELATIONS ON BI-PERIODIC JACOBSTHAL SEQUENCE Pethe, S.P., Phdte, C.N., A generliztion of the Fiboncci sequence, Applictions of Fiboncci numbers 5, , St. Andrews, Hordm, A.F., Jcobsthl representtion numbers, The Fiboncci Qurterly. 37, 40 54, Edson, M., Yyenie, O., A new generliztion of Fiboncci sequences nd the extended Binet s formul, INTEGERS Electron. J. Comb. Number Theor. 9, , Yyenie, O., A note on generlized Fiboncci sequence, Appl. Mth. Comput. 17, , Bilgici, G., Two generliztions of Lucs sequence, Applied Mthemtics nd Computtion 45, , Cosun,A., Tsr, N., The mtrix sequence in terms of bi-periodic Fiboncci numbers, rxiv: v mth.nt, 4 Apr Cosun,A., Yilmz, N., Tsr, A note on the bi-periodic Fiboncci nd Lucs mtrix sequences, rxiv: v1 mth.nt, 4 Apr Uygun, S., Owusu, E., A New Generliztion of Jcobsthl Numbers Bi-Periodic Jcobsthl Sequences, Journl of Mthemticl Anlysis 7 5, 8 39, Uygun, S., Owusu, E., Mtrix Representtion of Bi-Periodic Jcobsthl, rxiv: v1 mth.co, Feb Jun, S.P., Choi, K.H, Some properties of the Generlized Fiboncci Sequence by Mtrix Methods, Koren J. Mth 4 4, Gzintep University Deprtment of Mthemtics 7310, Sehitmil, Gzintep, Turey E-mil ddress: suygun@gntep.edu.tr