On quaternions with generalized Fibonacci and Lucas number components

Transcription

1 Polatl Kesm Advaces Dfferece Equatos (205) 205:69 DOI 0.86/s x R E S E A R C H Ope Access O quateros wth geeralzed Fboacc Lucas umber compoets Emrah Polatl * Seyhu Kesm * Correspodece: emrah.polatl@beu.edu.tr Departmet of Mathematcs, Bulet Ecevt Uversty, Zoguldak, 6700, Turkey Abstract I ths paper, we gve the expoetal geeratg fuctos for the geeralzed Fboacc geeralzed Lucas quateros, respectvely. Moreover, we gve some ew formulas for bomal sums of these quateros by usg ther Bet forms. MSC: B39 Keywords: geeralzed Fboacc quateros; geeralzed Lucas quateros Itroducto The well-kow Fboacc Lucas sequeces are defed by the followg recurrece relatos: for 0, F +2 F + + F L +2 L + + L, where F 0 0,F,L 0 2,L,respectvely.Here,F s the th Fboacc umber L s the th Lucas umber. The Bet formulas for the Fboacc Lucas sequeces are gve by F α β α β L α + β, where α β 5. 2 The umerous relatos coectg Fboacc Lucas umbers have bee gve by Vajda []. 205 Polatl Kesm. Ths artcle s dstrbuted uder the terms of the Creatve Commos Attrbuto 4.0 Iteratoal Lcese ( whch permts urestrcted use, dstrbuto, reproducto ay medum, provded you gve approprate credt to the orgal author(s) the source, provde a lk to the Creatve Commos lcese, dcate f chages were made.

2 Polatl Kesm Advaces Dfferece Equatos (205) 205:69 Page 2 of 8 Carltz, Fers, Layma Hoggatt studed may propertes of bomal sums of these umbers (see [2 5]). The geeralzed Fboacc Lucas sequeces, U } V },aredefedbythefollowg recurrece relatos: for 0 ay ozero teger p, U +2 pu + + U V +2 pv + + V, where U 0 0,U,V 0 2,V p, respectvely.ifwetakep,theu F (th Fboacc umber) V L (th Lucas umber). Let λ μ be the roots of the characterstc equato x 2 px 0. The the Bet formulas for the sequeces U } V } are gve by U λ μ V λ + μ, where λ (p + p 2 +4)/2μ (p p 2 +4)/2. From these equaltes, by puttg p 2 +4,weseethat+λ 2 λ + μ 2 μ. The geeratg fucto the expoetal geeratg fucto of a sequece a } are defed by a(x) â(x) a x a x!, respectvely. Now, we would lke to expla how the geeratg fucto of a gve sequece a } s derved: Frstly, we wrte dow a recurrece relato whch s a sgle equato expressg a terms of other elemets of the sequece. Ths equato should be vald for all tegers, assumg that a a 2 0. The we multply both sdes of the equato by x sum over all. Ths gves, o the left, the sum a x, whch s the geeratg fucto a(x). The rght-h sde should be mapulated so that t becomes some other expresso volvg a(x). Lastly, fwesolvetheresultgequato, wegetaclosedformfora(x). For more detals propertes related to the geeratg fuctos specal fuctos, we referto[6 2].

3 Polatl Kesm Advaces Dfferece Equatos (205) 205:69 Page 3 of 8 Aquaterop wth real compoets a 0, a, a 2, a 3,bass,, j, k s a elemet of the form p a 0 + a + a 2 j + a 3 k (a 0, a, a 2, a 3 ) (a 0 a 0 ), where 2 j 2 k 2, j k j, jk kj, k j k. Throughout ths paper, we work the real dvso quatero algebra wthout specfc refereces beg gve. The th Fboacc the th Lucas quateros were defed by Horadam [3]as Q F + F + + F +2 j + F +3 k K L + L + + L +2 j + L +3 k, respectvely, where F L are the th Fboacc umber the th Lucas umber. Smlar to Horadam, the th geeralzed Fboacc quatero th geeralzed Lucas quatero are defed by Q U + U + + U +2 j + U +3 k K V + V + + V +2 j + V +3 k, respectvely [4]. Here, U V are the th geeralzed Fboacc umber the th geeralzed Lucas umber. For 0, the followg recurrece relatos hold: Q +2 pq + + Q K +2 pk + + K, where Q 0 ( 0,, p, p 2 + ), Q (, p, p 2 +,p 3 +2p ), K 0 ( 2, p, p 2 +2,p 3 +3p ), K ( p, p 2 +2,p 3 +3p, p 4 +4p 2 +2 ).

4 Polatl Kesm Advaces Dfferece Equatos (205) 205:69 Page 4 of 8 I [5], Iak gave the Bet formulas for the geeralzed Fboacc geeralzed Lucas quateros as follows: Q λλ μμ K λλ + μμ, where λ +λ + λ 2 j + λ 3 k μ +μ + μ 2 j + μ 3 k. Iyer [6] gave some relatos coectg the Fboacc Lucas quateros. I [7], Swamy derved the relatos of geeralzed Fboacc quateros. Iak [8, 9] troduced the cocept of a hgher order quatero geeralzed quateros wth quatero compoets. Horadam [20] studed the quatero recurrece relatos. I [2], Halc derved geeratg fuctos may dettes for Fboacc Lucas quateros. I [22], Flaut Shpakvsky vestgated some propertes of geeralzed Fboacc quateros Fboacc-Narayaa quateros. Very recetly, Ramrez [4] has obtaed some combatoral propertes of the k-fboacc the k-lucas quateros. Ispred by these results, the preset paper we gve the expoetal geeratg fuctos for the geeralzed Fboacc geeralzed Lucas quateros. Moreover, we derve some ew formulas for bomal sums of these quateros by usg ther Bet forms. 2 Some propertes of geeralzed Fboacc Lucas quateros Theorem 2. ([4]) The geeratg fuctos for the geeralzed Fboacc geeralzed Lucas quateros are F(t) Q 0 +(Q pq 0 )t pt t 2 L(t) K 0 +(K pk 0 )t pt t 2, respectvely. Proof Let F(t) Q t L(t) K t.thewegetthefollowgequato: Q t pt Q t t 2 Q 0 +(Q pq 0 )t + Q t (Q pq Q 2 )t. 2 Sce, for each 2, the coeffcet of t s zero the rght-h sde of ths equato, we obta F(t) Q 0 +(Q pq 0 )t pt t 2.

5 Polatl Kesm Advaces Dfferece Equatos (205) 205:69 Page 5 of 8 Smlarly, we obta L(t) K 0 +(K pk 0 )t pt t 2. Theorem 2.2 The expoetal geeratg fuctos for the geeralzed Fboacc geeralzed Lucas quateros are Q! t λeλt μe μt K! t λe λt + μe μt. Proof By the Bet formula for the geeralzed Fboacc quateros, we get Q! t ( λλ μμ ) t λ (λt) λeλt μe μt.!! μ (μt) Smlarly, by the Bet formula for the geeralzed Lucas quateros we obta K (! t λλ + μμ ) t! (λt) (μt) λ + μ!!! λe λt + μe μt. Now, we gve the followg theorems whch the frst formulas are proved, the remag formulas ca be obtaed smlarly. Theorem 2.3 For, k 0, we have ( ) 2 Q +k f s eve, Q 2+k 2 K +k f s odd, K 2+k 2 K +k + f s eve, 2 Q +k f s odd, ( ) Q 2+k ( p) Q +k, ( ) K 2+k ( p) K +k.

6 Polatl Kesm Advaces Dfferece Equatos (205) 205:69 Page 6 of 8 Proof If we use the Bet formula for the geeralzed Fboacc quateros, we get Q 2+k λλk ( )( λλ 2+k μμ 2+k ( )λ 2 μμk ) λλk ( +λ 2 ) μμ k ( +μ 2 ) μ 2 λλk (λ ) μμk ( μ ) 2 Q +k f s eve, 2 K +k f s odd. Theorem 2.4 ([4]) For 0, we have p Q Q 2, p K K 2. Proof By the Bet formula for the geeralzed Fboacc quateros, we have p Q ( ( λλ )p μμ ) λ Theorem 2.5 For 0, we have (pλ) μ λ ( + pλ) μ ( + pμ) ( λλ 2 μμ 2) (pμ) Q 2. ( ) Q 4 2 ( 2 + p )Q f s eve, 2 ( 2 K 2 p Q ) f s odd, ( ) K 4 2 ( 2 + p )K f s eve, + 2 ( 2 Q p K ) f s odd, 2 Q K ( 2 p )Q f s eve, ( 2 K 2 + p Q ) f s odd, 2 ( 2 p )K f s eve, + ( 2 Q 2 + p K ) f s odd.

7 Polatl Kesm Advaces Dfferece Equatos (205) 205:69 Page 7 of 8 Proof By Theorem 2.3,weobta Q (+( ) ) Q 2 [ ( ] Q 2 + )( ) Q ( 2 + p )Q f s eve, ( 2 K 2 p Q ) f s odd. Theorem 2.6 For 0, we have 2 (Q ) 2 2 (2K (V 5 + p)v + (V 4 + V 0 )V ) f s eve, 2 (2Q (V 5 + p)u + (V 4 + V 0 )U ) f s odd, (K ) 2 2 (2K (V 5 + p)v + (V 4 + V 0 )V ) + f s eve, 2 (2Q (V 5 + p)u + (V 4 + V 0 )U ) f s odd. Proof From the quatero multplcato, we have (λ) 2 2λ ( (V 5 + p)λ + V 4 + V 0 ), (μ) 2 2μ ( (V 5 + p)μ + V 4 + V 0 ). If we cosder the Bet formula for the geeralzed Fboacc quateros, we have ( )( λλ μμ ) 2 (Q ) 2 (λ)2 (λ)2 [(λ) 2 λ 2 +(μ) 2 μ 2 2λμ(λμ) ] [(λ) 2 λ 2 +(μ) 2 μ 2] 2λμ ( )λ 2 + (μ)2 ( +λ 2 ) (μ) 2 ( + +μ 2 ) (λ)2 (λ ) + (μ)2 ( μ ) ( (λ) 2 λ +(μ) 2 ( μ) ) μ 2 ( ) (2K (V 5 + p)v + (V 4 + V 0 )V ) f s eve, 2 (2Q (V 5 + p)u + (V 4 + V 0 )U ) f s odd. Note that, by takg p Theorem2.6,weobta ( ) (Q ) (2K L 4L +4 ) f s eve, 5 2 (2Q F 4F +4 ) f s odd

8 Polatl Kesm Advaces Dfferece Equatos (205) 205:69 Page 8 of 8 ( ) (K ) (2K L 4L +4 ) f s eve, (2Q F 4F +4 ) f s odd, where Q K are the th Fboacc quatero the th Lucas quatero, respectvely. Competg terests The authors declare that they have o competg terests. Authors cotrbutos All authors typed, read, approved the fal mauscrpt. Ackowledgemets The authors would lke to thak the referees for ther valuable suggestos commets. Receved: 6 March 205 Accepted: 8 May 205 Refereces. Vajda, S: Fboacc Lucas Numbers the Golde Secto. Ells Horwood, Chchester (989) 2. Carltz, L: Some classes of Fboacc sums. Fboacc Q. 6, (978) 3. Carltz, L, Fers, HH: Some Fboacc Lucas dettes. Fboacc Q. 8, 6-73 (970) 4. Hoggatt, VE: Some specal Fboacc Lucas geeratg fuctos. Fboacc Q. 9, 2-33 (97) 5. Layma, JW: Certa geeral bomal-fboacc sums. Fboacc Q. 5, (977) 6. Adrews, GE, Askey, R, Roy, R: Specal Fuctos. Ecyclopeda of Mathematcs Its Applcatos, vol. 7. Cambrdge Uversty Press, Cambrdge (999) 7. Arac, S, Agyuz, E, Ackgoz, M: O a q-aalog of some umbers polyomals. J. Iequal. Appl. 205, 9 (205) 8. Graham, RL, Kuth, DE, Patashk, O: Cocrete Mathematcs. Addso-Wesley, Readg (994) 9. Klc, E: Sums of the squares of terms of sequece u }. Proc. Ida Acad. Sc. Math. Sc. 8, 27-4 (2008) 0. Km, T, Dolgy, DV, Km, DS, Rm, S-H: A ote o the dettes of specal polyomals. Ars Comb. 3A, (204). Smsek, Y: Specal fuctos related to Dedekd-type DC-sums ther applcatos. Russ. J. Math. Phys. 7, (200) 2. Srvastava, HM, Cho, J: Zeta q-zeta Fuctos Assocated Seres Itegrals. Elsever, Amsterdam (202) 3. Horadam, AF: Complex Fboacc umbers Fboacc quateros. Am. Math. Mo. 70, (963) 4. Ramrez, J: Some combatoral propertes of the k-fboacc the k-lucas quateros. A. Ştţ. Uv. Ovdus Costaţa, Ser. Mat. 23(2),20-22 (205) 5. Iak, AL: Exteded Bet forms for geeralzed quateros of hgher order. Fboacc Q. 9, (98) 6. Iyer, MR: A ote o Fboacc quateros. Fboacc Q. 3, (969) 7. Swamy, MNS: O geeralzed Fboacc quateros. Fboacc Q. 5, (973) 8. Iak, AL: Geeralzed quateros of hgher order. Fboacc Q. 5, (977) 9. Iak, AL: Geeralzed quateros wth quatero compoets. Fboacc Q. 5, (977) 20. Horadam, AF: Quatero recurrece relatos. Ulam Q. 2, (993) 2. Halc, S: O Fboacc quateros. Adv. Appl. Clfford Algebras 22, (202) 22. Flaut, C, Shpakvsky, V: O geeralzed Fboacc quateros Fboacc-Narayaa quateros. Adv. Appl. Clfford Algebras 23, (203)