GAMALIEL CERDA-MORALES 1. Blanco Viel 596, Valparaíso, Chile. s: /

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1 THE GELIN-CESÀRO IDENTITY IN SOME THIRD-ORDER JACOBSTHAL SEQUENCES arxiv: v1 [math.co] 20 Oct 2018 GAMALIEL CERDA-MORALES 1 1 Istituto de Matemáticas Potificia Uiversidad Católica de Valparaíso Blaco Viel 596 Valparaíso Chile. s: gamaliel.cerda@usm.cl / gamaliel.cerda.m@mail.pucv.cl Abstract. I this paper we deal with two families of third-order Jacobsthal sequeces. The first family cosists of geeralizatios of the Jacobsthal sequece. We show that the Geli-Cesàro idetity is satisfied. Also we defie a family of geeralized third-order Jacobsthal sequeces {J 3 } 0 by the recurrece relatio J 3 +3 J3 +2 +J J3 0 with iitials coditios J 3 0 a J 3 1 b ad J 3 2 c where a b ad c are o-zero real umbers. May sequeces i the literature are special cases of this sequece. We fid the geeratig fuctio ad Biet s formula of the sequece. The we show that the Cassii ad Geli-Cesàro idetities are satisfied by the idices of this geeralized sequece. Mathematical subject classificatio: 05A15 11B39. Key words: Third-order Jacobsthal sequece geeratig fuctio Jacobsthal sequece geeralized third-order Jacobsthal sequece. 1. Itroductio ad Prelimiaries The Jacobsthal umbers have may iterestig properties ad applicatios i may fields of sciece see e.g. [Ba Ho2 Ho3]. The Jacobsthal umbers J are defied by the recurrece relatio 1.1 J 0 0 J 1 1 J +2 J +1 +2J 0. Aother importat sequece is the Jacobsthal-Lucas sequece. This sequece is defied by the recurrece relatio j +2 j +1 +2j where j 0 2 ad j 1 1 see [Ho3]. I [Cook-Bac] the Jacobsthal recurrece relatio is exteded to higher order recurrece relatios ad the basic list of idetities provided by A. F. Horadam [Ho3] is expaded ad exteded to several idetities for some of the higher order cases. For example the third-order Jacobsthal umbers {J 3 } 0 ad thirdorder Jacobsthal-Lucas umbers {j 3 } 0 are defied by 1.2 J 3 +3 J3 +2 +J J3 J3 0 0 J 3 1 J

2 2 G. CERDA-MORALES ad 1.3 j 3 +3 j3 +2 +j j3 j j j respectively. Some of the followig properties give for third-order Jacobsthal umbers ad third-order Jacobsthal-Lucas umbers are used i this paper for more details see [Ce Ce1 Cook-Bac]. Note that Eqs. 1.8 ad 1.12 have bee corrected i this paper sice they have bee wrogly described i [Cook-Bac] J 3 +j j 3 3J3 2j3 3 3 { 1.6 J J3 1. j 3 4J3 2 if 1 mod 3 1 if 1 mod 3 2 if 0 mod 3 3 if 1 mod 3 1 if 2 mod j j3 3J j 3 J ad 1.12 J 3 k j 3 1 if 0 mod 3 1 if 1 mod 3 0 if 2 mod 3 2 j J 3 j3 4 { J 3 +1 if 0 mod 3 1 if 0 mod J 3 +1 J j Usig stadard techiques for solvig recurrece relatios the auxiliary equatio ad its roots are give by x 3 x 2 x2 0; x 2 ad x 1±i 3. 2 Note that the latter two are the complex cojugate cube roots of uity. Call them ω 1 ad ω 2 respectively. Thus the Biet formulas ca be writte as 1.13 J i 3 ω1 32i 3 ω

3 THE GELIN-CESÀRO IDENTITY IN THIRD-ORDER JACOBSTHAL SEQUENCES 3 ad 1.14 j i 3 ω1 + 32i 3 ω 2 respectively. Now we use the otatio 1.15 V 2 Aω 1 Bω 2 ω 1 ω 2 2 if 0 mod 3 3 if 1 mod 3 1 if 2 mod 3 where A 32ω 2 ad B 32ω 1. Furthermore ote that for all 0 we have 1.16 V V +1 V2 V ad V From the Biet formulas ad Eq we have 1.1 J V 2 ad j V 2 O the other had the Geli-Cesàro idetity [Di p. 401] states that 1.18 F 4 F 2 F 1 F +1 F +2 1 where F is the classic -th Fiboacci umber. Furthermore Melham ad Shao [Me-Sha] obtaied geeralizatios of the Geli-Cesàro idetity. Recetly Sahi [Sa] showed that the Geli-Cesàro idetity is satisfied for two families of coditioal sequeces. Motivated by [Me-Sha Sa] i this paper we deal with two families of thirdorder Jacobsthal sequeces. The first family cosists of the sequeces deoted by {J 3 } ad studied i [Ce Cook-Bac]. We show that the Geli-Cesàro idetity is satisfied by the sequece {J 3 }. Also we defie a family of geeralized thirdorder Jacobsthal sequeces {J 3 } by the recurrece relatio J 3 +3 J J J3 0 with iitials coditios J 3 0 a J 3 1 b ad J 3 2 c where a b ad c are o-zero real umbers. May sequeces i the literature are special cases of this geeralized sequece. We fid the geeratig fuctio ad Biet formula for the sequece {J 3 } 0. The we show that Catala ad Geli-Cesàro idetities are satisfied by this geeralized sequece.. 2. The first family of third-order Jacobsthal sequeces Recetly the authors itroduced i [Cook-Bac] a further geeralizatio of the Jacobsthal sequece amely the third-order Jacobsthal sequece defied by Eq The

4 4 G. CERDA-MORALES Lemma 2.1 Catala Idetity for J 3. For ay oegative itegers ad r we have 2.1 J 3 2 J 3 rj 3 +r 1 where U 2 r j 3 r1 J3 r r V 2 r Proof. From Eqs ad 1.1 we obtai 2 3 J J r J3 +r { 2 +1 V 2 1 { r+1 V 2 r V 2 + 2V 2 U 2 r +2 r V 2 +r 2 } 2 +r+1 V 2 +r V r+1 V 2 +r +2+r+1 V r V 2 2 2V +2 r V 2 + r V 2 2 V 2 rv 2 +r +r r V 2 2 r 2V +2 r V 2 +r 2 r V2 r V 2 +r + U 2 r } 2 where U r 2 j 3 r1 J3 r+1 usig Eq The proof is completed. Theorem 2.2 Geli-Cesàro Idetity. For ay o-egative itegers 2 we have 2.2 J 3 4 J J3 1 J3 +1 J3 +2 where W J R R2 +1 5V +1 3V 2 3R R R 2 +1 R2 +2 ad V 2 as i Eq Proof. For r 1 ad r 2 ad Eq we get respectively 2 J 3 3 J 1 J V V +2 1 V V V +V V 2 3V W 2 +1 U 2 1 2

5 THE GELIN-CESÀRO IDENTITY IN THIRD-ORDER JACOBSTHAL SEQUENCES 5 ad J 3 2 J 3 2 J V V +2 2 V V V +V V W V 2 U by usig Lemma 2.1 ad the property V 2 V 2 +3 for all 0. Note that the sequece W 2 satisfies relatio 5V V W So we obtai J 3 2 J3 1 J3 +1 J3 +2 J 3 J J 3 12 W 2 +1 The we have 4 3 J J 3 12 W J W W W J3 1 J3 +1 J W J W W W W W 2 +1 W2 +2 J if 0 mod 3 1 J if 1 mod J if 2 mod 3 1 The proof is completed. 3. The secod family of geeralized third-order Jacobsthal sequeces Here we defie a ew geeralizatio of the third-order Jacobsthal sequece {J 3 } 0. Let us deote this sequece by {J 3 } 0 which is defied recursively

6 6 G. CERDA-MORALES by 3.1 { J 3 +3 J3 +2 +J J3 0 J 3 0 a J 3 1 b J 3 2 c where a b ad c are real umbers. For example the first seve terms of the sequece are {a b c 2a+b+c 2a+3b+2c 4a+4b+5c 10a+9b+9c}. For a 0 ad b c 1 we get the ordiary third-order Jacobsthal sequece. Also whe a 2 b 1 ad c 5 we get the third-order Jacobsthal-Lucas sequece which is defied i [Cook-Bac]. I this study first we obtai the geeratig fuctio ad the Biet s formula for the sequece {J 3 } 0. Fially we show some properties for example the Catala ad Geli-Cesàro idetity are satisfied by this sequece. Now we ca give the geeratig fuctio of the sequece. Theorem 3.1 Geeratig fuctio. The geeratig fuctio for the sequece {J 3 } 0 is 3.2 F J t a+bat+cbat2 1tt 2 2t 3. Proof. Let F J t J 3 0 +J 3 1 t+j3 2 t2 + 0 J3 t which is the formal power series of the geeratig fuctio for {J 3 }. We obtai that 1tt 2 2t 3 F J t J 3 0 +J 3 1 t+j3 2 t2 + J 3 0 tj3 1 t2 J 3 2 t3 J 3 0 t2 J 3 1 t3 J 3 2 t4 2J 3 0 t3 2J 3 1 t4 2J 3 2 t5 J 3 0 +J 3 1 J 3 0 t+j3 2 J 3 1 J 3 0 t2 sice J 3 +3 J3 +2 +J J3 0 ad the coefficiets of t for 3 are equal with zero. The the theorem is proved. I fact we ca give Biet s formula for the sequece as follows. Theorem 3.2. For 0 we have J 3 c+b+a 2ω 1 2ω c2+ω1 b+2ω 1 a + 2ω 2 ω 1 ω 2 2 c2+ω2 b+2ω 2 a 2ω 1 ω 1 ω 2 ω2. ω 1

7 THE GELIN-CESÀRO IDENTITY IN THIRD-ORDER JACOBSTHAL SEQUENCES Proof. The solutio of Eq. 3.1 is 3.4 J 3 A J 2 +B J ω 1 +C J ω 2. The let J 3 0 A J + B J + C J J 3 1 2A J + B J ω 1 + C J ω 2 ad J 3 2 4A J + B J ω1 2 +C J ω2. 2 Therefore we have 2ω 1 2ω 2 A J cω 1 +ω 2 b+ω 1 ω 2 a 2ω 1 ω 1 ω 2 B J c2+ω 2 b+2ω 2 a2ω 2 ω 1 ω 2 C J c2+ω 1 b+2ω 1 a. Usig A J B J ad C J i Eq. 3.4 we obtai c+b+a 2ω 1 2ω 2 J 3 + The proof is completed. 2 c2+ω1 b+2ω 1 a 2ω 2 ω 1 ω 2 Theorem 3.3. Assume that x 0. We obtai 3.5 J 3 k x k 1 x νx where νx x 3 x 2 x2. 2J 3 + { x +1 c2+ω2 b+2ω 2 a 2ω 1 ω 1 ω 2 ω2. ω 1 J 3 +2 J3 +1 x+j 3 +1 x2 } cbaabx+ax 2 Proof. From Theorem 3.2 we have J 3 k x k c+b+a 2 2ω 1 2ω 2 x c2+ω2 b+2ω 2 a 2ω 1 ω 1 ω 2 c2+ω1 b+2ω 1 a + 2ω 2 ω 1 ω 2 k ω1 k x ω2 k. x By cosiderig the defiitio of a geometric sequece we get J 3 k x k c+b+a 2 +1 x +1 2ω 1 2ω 2 x 2x c2+ω2 b+2ω 2 a ω +1 1 x +1 2ω 1 ω 1 ω 2 x ω 1 x c2+ω1 b+2ω 1 a ω +1 2 x ω 2 ω 1 ω 2 x ω 2 x 1 A J 2 +1 x +1 ω 1 xω 2 x x B J ω1 +1 x νx +1 2xω 2 x +C J ω2 +1 x +1 2xω 1 x

8 8 G. CERDA-MORALES where 3.6 { A J c+b+a 2ω B 12ω 2 J c2+ω2b+2ω2a 2ω C 1ω 1ω 2 J c2+ω1b+2ω1a 2ω 2ω 1ω 2 ad νx x 3 x 2 x2. Usig ω 1 +ω 2 1 ad ω 1 ω 2 1 if we rearrage the last equality the we obtai A J 2 +1 x +1 1+x+x 2 J 3 k x k 1 x νx 1 x νx B J ω1 +1 x +1 2ω 2 2+ω 2 x+x 2 +C J ω2 +1 x +1 2ω 1 2+ω 1 x+x 2 A J x+x 2 B J ω1 +1 2ω 2 2+ω 2 x+x 2 +C J ω2 +1 2ω 1 2+ω 1 x+x 2 A J 1+x+x 2. x +1 B J 2ω 2 2+ω 2 x+x 2 +C J 2ω 1 2+ω 1 x+x 2 So the proof is completed. I the followig theorem we give the sum of geeralized third-order Jacobsthal sequece correspodig to differet idices. Theorem 3.4. For r m we have 3. J 3 mk+r 1 σ J 3 m+1+r J3 r +2 m J 3 m+r 2m J 3 rm J 3 m+1+r µm+j3 r µm +J 3 m+2+r J3 r+m where σ 2 m m ω m 1 +ωm 2 2 ad µm 2m +ω m 1 +ωm 2. Proof. Let us take A J B J ad C J i Eq The we write J 3 mk+r A J2 r 2 mk B J ω1 r ω1 mk +C J ω2 r ω2 mk 2 A J 2 r m m B J ω1 r ω m ω1 m 1 +C J ω2 r ω m ω2 m 1 A 1 J 2 m+1+r 2 r ω 1 m ω2 m ω1 m +ω2 m +1 B J ω m+1+r 1 ω1 r 2 m ω2 m σ 2m +ω2 m +1 +C J ω m+1+r 2 ω2 r 2 m ω1 m 2m +ω1 m+1

9 THE GELIN-CESÀRO IDENTITY IN THIRD-ORDER JACOBSTHAL SEQUENCES 9 where σ 2 m m ω1 m +ωm 2 2. After some algebra we obtai J 3 m+1+r J3 r +2 m J 3 m+r 2m J 3 rm J 3 m+1+r µm+j3 r µm J 3 mk+r 1 σ +J 3 m+2+r J3 r+m where µm 2 m +ω1 m +ωm 2. The proof is completed. 4. Mai results We use the ext otatio for the Biet formula of geeralized third-order Jacobsthal sequece J 3. Let 4.1 V 2 Aω 1 Bω 2 ω 1 ω 2 c+b6a if 0 mod 3 2c5b+2a if 1 mod 3 3c+4b+4a if 2 mod 3 where A 2ω 2 c2+ω 2 b+2ω 2 a ad B 2ω 1 c2+ω 1 b+2ω 1 a. Furthermore ote that for all 0 we have 4.2 V 2 +2 V2 +1 V2 V 2 0 c+b6a V 2 1 2c5b+2a. From the Biet formula 3.3 ad Eq. 4.1 we have 4.3 J 3 1 ρ2 V 2 where ρ a+b+c. I particular if a 0 ad b c 1 we obtai J 3 J 3. Theorem 4.1 Catala Idetity for J 3. For ay oegative itegers ad r we have 4.4 J 3 2J 3 r J3 +r 1 where U r 2 j 3 r1 J3 r+1 ad ρ a+b+c. Proof. From Eq. 4.3 we obtai 2 3 J J r J3 +r { ρ2 V 2 2 ρ 2 r V 2 r 2V2 +2 r V 2 +r 2 +4a 2 +3b 2 +c 2 2ac3bc U 2 r 2 } ρ2 r V 2 r ρ2 +r V 2 +r ρ ρv 2 + V 2 ρ r ρv 2 2 ρ 2 r V 2 r 2V2 +2 r V 2 +r V V 2 2 +r +2+r ρv 2 r V2 r V2 +r r V2 +r.

10 10 G. CERDA-MORALES After some algebra we obtai 2 J 3 3 J r J3 +r 1 2 ρ 2 r V 2 r 2V 2 +2 r V 2 +r 2 +4a 2 +3b 2 +c 2 2ac3bc U 2 r where U r 2 j 3 r1 J3 r+1 usig Eq The proof is completed. Theorem 4.2 Geli-Cesàro Idetity. For ay o-egative itegers 2 we have 4.5 J J J3 1 J3 +1 J3 +2 J 3 w ρω where W V +1 3V 2 2 2ω+2 2 ρ 3W W2 +1 3W W ρ 2 W 2 +1 W2 +2 ad V 2 as i Eq Proof. From Eq. 4.4 i Theorem 4.1 ad r 1 ad r 2 we obtai { } 2 J 3 3 J 1 J ρ 2V 2 1 2V V a 2 +3b 2 +c 2 2ac3bc { } ρ 4V 2 1 4V2 +V a 2 +3b 2 +c 2 2ac3bc { } 1 4a 2 +3b 2 +c 2 2ac3bc 2 1 ρ 5V 2 3V ω 2 1 ρw 2 +1 ad J 3 2 J 3 2 J ω ρw 2 +2 where 5V 2 3V 2 1 W2 +1 ad ω 4a2 +3b 2 +c 2 2ac3bc. So we ca write the ext equality: J 3 2 J3 1 J3 +1 J3 +2 J J ω 2 1 ρw J 3 ω 2 1 ρw 2 +1 J 3 2ω ρw ρw 2 +1 ω ρw 2 +2 ω ρw 2 +2.

11 THE GELIN-CESÀRO IDENTITY IN THIRD-ORDER JACOBSTHAL SEQUENCES 11 The we have 4 3 J J J3 1 J3 +1 J3 +2 J 3 w ρω The proof is completed. 2 2ω+2 2 ρ 3W W2 +1 3W W ρ 2 W 2 +1 W2 +2 By the aid of the last theorem we have the followig corollary.. Corollary 4.3. For ay o-egative itegers 2 we have 4 3 J J J3 1 J3 +1 J ω J ρa if 0 mod 3 w ρωa ρ 2 T 2 22ω J ρb if 1 mod 3 w ρωb ρ 2 T 2 22ω J ρc if 2 mod 3 w ρωc ρ 2 T 2 where { A c10b+24a B 11c+23b2a C 12c13b22a ad T 2 2cb6ac4b+4a if 0 mod 3 c4b+4a3c+5b+2a if 1 mod 3 3c+5b+2a2cb6a if 2 mod 3 Refereces. [Ba] P. Barry Triagle geometry ad Jacobsthal umbers Irish Math. Soc. Bull [Ce] G. Cerda-Morales Idetities for Third Order Jacobsthal Quaterios Advaces i Applied Clifford Algebras [Ce1] G. Cerda-Morales O a Geeralizatio of Triboacci Quaterios Mediterraea Joural of Mathematics 14: [Cook-Bac] C. K. Cook ad M. R. Baco Some idetities for Jacobsthal ad Jacobsthal- Lucas umbers satisfyig higher order recurrece relatios Aales Mathematicae et Iformaticae [Di] L. E. Dickso History of the Theory of Numbers Vol. I Chelsea Publishig Co. New York [Ho2] A. F. Horadam Jacobsthal ad Pell Curves The Fiboacci Quarterly

12 12 G. CERDA-MORALES [Ho3] A. F. Horadam Jacobsthal represetatio umbers The Fiboacci Quarterly [Me-Sha] R. S. Melham ad A. G. Shao A geeralizatio of the Catala idetity ad some cosequeces The Fiboacci Quarterly [Sa] M. Sahi The Geli-Cesàro idetity i some coditioal sequeces Hacettepe Joural of Mathematics ad Statistics