#A27 INTEGERS 3 (203) SOME WEIGHTED SUMS OF PRODUCTS OF LUCAS SEQUENCES Emrah Kılıç Mathematcs Departmet, TOBB Uversty of Ecoomcs ad Techology, Akara, Turkey eklc@etu.edu.tr Neşe Ömür Mathematcs Departmet, Kocael Uversty, İzmt Kocael, Turkey eseomur@kocael.edu.tr Receved: 8/6/2, Accepted: 4/2/3, Publshed: 5/0/3 Abstract I ths paper, we cosder the weghted sums of products of Lucas sequeces of the form r mk s m(tk), k k0 where r ad s are the terms of Lucas sequeces {U } ad {V } for some postve tegers t ad m. By usg geeratg fucto methods, we compute the weghted sums of products of Lucas sequeces ad show that these sums could be expressed va terms of the Lucas sequeces.. Itroducto Defe secod order lear recurreces {U } ad {V } for > 0 as U pu U 2, V pv V 2, where U 0 0, U ad V 0 2, V p, respectvely. If p, the U F (th Fboacc umber) ad V L (th Lucas umber). The Bet formulas of {U } ad {V } are where α, β U ad V α β, p ± /2 ad p 2 4.
INTEGERS: 3 (203) 2 Let A (x) ad B (x) be the expoetal geeratg fuctos of sequeces {a } ad {b }, that s, A (x) 0 a x! ad B (x) 0 k0 b x!. The the covoluto of them s gve by A (x) B (x) x a k b k k!. 0 May authors have cosdered ad computed may kds of bomal sums as well as weghted bomal sums wth terms of certa umber sequeces. As a cosequece of covoluto of two expoetal geeratg fuctos, we have the followg results from the lterature (see [2]): F m L m m, F m F m m, (.) 0 0 L m F m m, 0 0 L m L m m. (.2) I ths paper, motvated by (.) ad (.2), we cosder the followg ew four kds of weghted bomal sums wth the product of terms of the sequeces {U } ad {V } : 0 0 U m V m(k), U m U m(k), 0 0 V m U m(k), V m V m(k), for some tegers k ad m. We cosder the sums above ad the show that the sums could cely be expressed terms of the terms of the sequeces {U } ad {V }. Because of the dces of the terms the sums, the covoluto of expoetal geeratg fuctos ca ot be used for computg these sums. Our approach for computg these kd of sums s maly to use geeratg fucto methods ad the Bet formula of the sequeces. For computg weghted bomal sums wth the product of terms of bary sequeces ad usg geeratg fuctos dervg combatoral dettes, we refer to [, 3]. 2. The Ma Results Frst we gve a useful auxlary lemma ad ts drect cosequeces. After ths we gve our ma results. By the Bet formulas of {U } ad {V }, we have the followg
INTEGERS: 3 (203) 3 results wthout proof: Lemma. For odd m, α 2m α m U m ad β 2m β m U m, ad for eve m, α 2m α m V m ad β 2m β m V m. As straghtforward cosequeces of Lemma, we have the followg results. Corollary. Let m be a oegatve odd teger. The for > 0, U 2m Um 2 V m f s odd, 2 U m f s eve. 0 Proof. By Lemma, we wrte for odd, α 2m α m U m 2 ad β 2m β m U m 2 or 0 ad so α 2m α m U m p 2 4 2 ad 0 α 2m β 2m 0 0 β 2m β m U m p 2 4 2 2 U m (α m β m ) U 2m 2 U m V m, as clamed. For eve, cosder α 2m α m U m 2 ad β 2m β m U m 2 or ad so as clamed. 0 α 2m α m U m 2 ad β 2m β m U m 2 0 0 α 2m β 2m 2 U m (α m β m ) U 2m 2 U m U m, 0
INTEGERS: 3 (203) 4 For example, for odd m > 0 ad > 0, we have 0 whch ca be foud [4]. F 2m Fm 5 2 L m f s odd, 5 2 F m f s eve, Corollary 2. Let m be a oegatve eve teger. The for > 0, 0 0 U 2m VmU m, V 2m VmV m. Proof. By Lemma, we have that for eve m, α 2m α m V m ad β 2m β m V m ad wrte α 2m α m V m ad β 2m β m V m, whch, by the bomal theorem, gves us 0 α 2m α m Vm ad 0 β 2m β m V m. By subtractg these two equaltes sde by sde ad the Bet formula of {U }, we obta U 2m V mu m. 0 By addg the above two equaltes ad the Bet formula of {V }, we obta as clamed. 0 V 2m V mv m, For eve m > 0 ad > 0, we obta 0 F 2m L mf m ad 0 L 2m L ml m.
INTEGERS: 3 (203) 5 Theorem. Let k be a oegatve teger. For odd m, U m V kmm 2 U U(k)m f s eve, m f s odd. For eve m, 0 0 V (k)m U m V kmm V mu m(k) 2 U km. (2.) Proof. Multplyg the left-had sde of (2.) by z ad summg over ad by the Bet formulas of {U } ad {V }, we derve for odd m, 0 z 0 0 0 0 0 U m V kmm α z km2m β km2m α km β km ( ) 0 0 α km z β 2m 0 0 α 2m 0 0 α km β km ( ) m z α m(k2) z β m(k2) z β km z ( (α km z)) ( (β km z)) ( ) m α km z ( (α km z)) ( ) m β km z 0 ( (β km z)) ( α km z) αm(k2) z ( β km z) α km z βm(k2) z β km z α km ( ( ) m ) z β km ( ( ) m ) z zα km ( α 2m ) zβ km ( β 2m ) α km α 2m β km β 2m z. 0 Therefore, we get the detty U m V kmm U 2mkm. 0 0 ( )
INTEGERS: 3 (203) 6 Usg Lemma, we wrte for eve, α km α 2m β km β 2m α km α m U m β km β m U m α km α m U m 2 β km β m Um 2 U m 2 U(k)m. O the other had, we get for odd α km α 2m β km β 2m α km α m U m β km β m U m α km α m Um 2 β km β m U m 2 p2 4 α km α m Um 2 β km β m Um 2 U m 2 V(k)m, as clamed. By combg the above two results, the proof s complete for the case m s odd. Now we cosder the case m s eve: 0 0 0 0 α km z β 2m 0 0 α 2m 0 0 α km β km z α m(k2) z β m(k2) z ( (α km z)) ( (β km z)) α km z ( (α km z)) β km z 0 ( (β km z)) ( α km z) αm(k2) z ( β km z) α km z βm(k2) z β km z 2α km z 2β km z β km z
INTEGERS: 3 (203) 7 zα km ( α 2m ) zβ km ( β 2m ) 2α km z 2β km z α km α 2m β km β 2m 2 α km β km z 0 By Lemma, we wrte as clamed. α km (α m V m ) β km (β m V m ) 2 α km β km Vm α m(k) β m(k) 2 α km β km VmU m(k) 2 U km, Theorem 2. Let k be a oegatve teger. For odd m, 0 V m V kmm 2 U V(k)m f s eve, m f s odd. U (k)m (2.2) For eve m, 0 V m V kmm V mv (k)m 2 V km. Proof. Multplyg the left-had sde of (2.2) by z ad summg over, we wrte V m V kmm z 0 0 α m β m α kmm β kmm z 0 0 α km2m β km2m ( ) m α km β km z 0 0 α α 2m km z β β 2m km z 0 0 0 0 ( ) m α km β km z 0 0 α km2m z β km2m z 0 ( α km z) ( β km z)
INTEGERS: 3 (203) 8 0 0 ( ) m α km z ( ) m 0 ( α km z) β km z ( β km z) zα km ( α 2m ) zβ km ( β 2m ) zα km ( ( ) m ) zβ km ( ( ) m ). If m s eve, the by Lemma, we wrte V m V kmm z whch gves us zα m(k) V m zβ m(k) V m 2zα km 2zβ km α m(k) β m(k) Vm 2 α km β km z 0 If m s odd, the 0 0 0 V m V kmm V mv (k)m 2 V km. V m V kmm z zα α km m U m p2 4 zβ β km m U m p2 4 α (k)m U m p2 4 β m(k) U m p2 4 z. 0 Now we cosder two cases: frst f s odd, the we obta V m V kmm z 0 0 0 0 0 0 α (k)m U m p 2 4 2 β m(k) U m p 2 4 2 Um 2 α (k)m β m(k) z Um 2 α (k)m β m(k) U m 2 U(k)m z. z z
INTEGERS: 3 (203) 9 Secod, f s eve, the we obta 0 0 V m V kmm z 0 0 α (k)m Um 2 β (k)m Um 2 z Um p 2 4 2 α (k)m β (k)m z 0 U m 2 V(k)m z. By combg the last two results, we prove the clam for odd m. Thus the proof s complete. Smlar to the proof methods of Theorems ad 2, we gve the followg results wthout proof. Theorem 3. Let k be a oegatve teger. For odd m, V m U kmm 2 U U(k)m f s eve, m f s odd. For eve m, 0 0 V (k)m V m U kmm V mu m(k) 2 U km Theorem 4. Let k be a oegatve teger. For odd m, U m U kmm 2 U V(k)m f s eve, m f s odd. 0 For eve m, 0 U (k)m U m U kmm V m V (k)m 2 V km. Refereces [] E. Kılıç, Y. Ulutas ad N. Omur, Formulas for weghted bomal sums wth the powers of terms of bary recurreces, Mskolc Math. Notes 3 () (202), 53-65. [2] T. Koshy, Fboacc ad Lucas umbers wth applcatos. Pure ad Appled Mathematcs, Wley-Iterscece, New York, 200. [3] P. Staca, Geeratg fuctos, weghted ad o-weghted sums for powers of secod-order recurrece sequeces, Fboacc Quarterly 4.4 (2003), 32-333. [4] S. Vajda, Fboacc & Lucas umbers, ad the golde secto. Joh Wley & Sos, Ic., New York, 989.