FORMULAS FOR BINOMIAL SUMS INCLUDING POWERS OF FIBONACCI AND LUCAS NUMBERS

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1 U.P.B. Sc. Bull., Seres A, Vol. 77, Iss. 4, 015 ISSN FORMULAS FOR BINOMIAL SUMS INCLUDING POWERS OF FIBONACCI AND LUCAS NUMBERS Erah KILIÇ 1, Iler AKKUS, Neşe ÖMÜR, Yücel Türer ULUTAŞ3 Recently Prodnger [] proved general expanson forulas for sus of powers of Fbonacc and Lucas nubers. In ths paper, we wll prove general expanson forulas for bnoal sus of powers of Fbonacc and Lucas nubers. Keywords: Fbonacc and Lucas nubers, bnoal sus, powers. MSC010: 11B39, 05A Introducton The Fbonacc nubers and Lucas nubers L n are defned by the followng recursons: for n > 0, and L n+1 L n + L n 1, where F 0 0, F 1 1 and L 0, L 1 1, respectvely. If the roots of the characterstc equaton x x 1 0 are α and β, then the Bnet forulas for the are αn β n α β and L n α n + β n. Weann and Cooper [4] entoned about soe conjectures of Melha for the su: F Oze [1] consdered Melha s su and then he gave an explct expanson for Melha s su as a polynoal n +1. In general, Prodnger [] derved the general forula for the su: F +ε +δ, where ε, δ {0, 1}, as well as the evaluatons of the correspondng sus for Lucas nubers. 1 TOBB Econocs and Technology Unversty, Matheatcs Departent, Anara, Turey, e-al: elc@etu.edu.tr Kırıale Unversty, Faculty of Arts and Scences, Departent of Matheatcs, Yahshan, Kırıale, Turey, e-al: aus.tr@gal.co 3 Kocael Unversty, Departent of Matheatcs, Izt, Turey, e-al: neseour@ocael.edu.tr, turery@ocael.edu.tr 69

2 70 Erah KILIÇ, Iler AKKUS, Neşe ÖMÜR, Yücel Türer ULUTAŞ In [5], the authors gave forulas for the alternatng analogues of sus of Melha for Fbonacc and Lucas nubers of the fors 1 F +ε +δ and 1 L +ε +δ, 1 where ε, δ {0, 1}. In ths paper, we consder certan bnoal sus gven by F +ε +δt, L +ε +δt, 1 F +ε +δt and 1 1 L +ε +δt, where t s an postve nteger and ε, δ {0, 1}. Throughout ths paper, we use the ndcator functon [S] defned by 1 f the stateent S s true or 0 otherwse. We recall soe facts for the readers convenence n [3]: For any real nubers and n, and + n t t/ 1 n t t n t + n t t + n t/ [t s even] 1 t/ Fro [6], we have the followng result: t/ 1 t n 1 t + n t t + n t/ 1 t/ [t s even]. t/ Lea 1.1. Let r and s be arbtrary ntegers. Then 5 n 1/ Fs n L sn+r f n s odd, F r+s 5 n/ Fs n F sn+r f n s even, L n s F sn+r f s s even. L r+s 1 F r+s 5 n+1/ Fs n F sn+r f n s odd, 5 n/ Fs n L sn+r f n s even, L n s L sn+r f s s even. 5 n/ Fs n F sn+r f n s even, 5 n 1/ Fs n L sn+r f n s odd, 1 n L n s F sn+r f s s odd. f s s odd, f s s odd, f s s even, 3 4 5

3 Forulas for Bnoal Sus Includng Powers of Fbonacc and Lucas Nubers 71 v 1 L r+s 5 n/ Fs n L sn+r f n s even, 5 n+1/ Fs n F sn+r f n s odd, 1 n L n s L sn+r f s s odd. f s s even, 6. Soe Bnoal Sus for Fbonacc Nubers Here we consder bnoal and alternatng bnoal sus of powers of Fbonacc nubers. Theore.1. For t > 0, Ft For odd t > 0, F +1 t 5 n 5 n 1 L n t L tn n F n +1t F +1tn f n s even, and for even t > 0, F +1 t F n +1t L +1tn f n s odd, L n +1t F +1tn Proof. Fro the Bnet forulas of { } and {L n }, and by, we wrte Ft n α t β t α β 1 1 n α β 1 α t + β t n L t + 1 n, whch, by 4 n Lea 1.1 and snce n n n, equvalent to n Ft L n t L tn + 1 n,

4 7 Erah KILIÇ, Iler AKKUS, Neşe ÖMÜR, Yücel Türer ULUTAŞ as claed. Consder F +1 t n α t β t +1 α β α β +1 1 α +1 t β +1 t n F +1 t, whch, by tang s + 1 t and r 0 n 3 n Lea 1.1, gves the claed results. Followng the proof way of Theore.1, we have the followng result wthout proof: Theore.. For t > 0, F+1t t+1 L n t L tn t+1 n. For odd t > 0, F +1 +1t 5 n / 5 n 1/ t F +1 tn+1 +1t L +1tn+1 f n s even, f n s odd, and, for even t > 0, F +1 +1t L n +1 t F +1 tn+1.

5 Forulas for Bnoal Sus Includng Powers of Fbonacc and Lucas Nubers 73 Theore.3. For t > 0, 1 F+1t 5 n 1 1 t+1 F t n L tn t+1 [n 0] 5 n t+1 For odd t > 0, 1 F +1 +1t 1n 5 and, for even t > 0, 1 F +1 +1t 5 n 5 n 1 t F tn+1 f n s even, f n s odd. + 1 L n +1 t F +1 tn+1, t F +1 tn+1 Proof. For t > 0, by, consder 1 F+1t t L +1 tn+1 α 1 +1t β +1t α β 1 1 n α β 1 1 t+1 α t+1 + β t α β t+1 1 t t 1 L t+1 1, f n s even, f n s odd.

6 74 Erah KILIÇ, Iler AKKUS, Neşe ÖMÜR, Yücel Türer ULUTAŞ whch, by tang s r t n 6 n Lea 1.1 and n n 1 [n 0], gves the claed result. For t > 0, by, consder n 1 F +1 +1t α 1 +1t β +1t α β 1 α β α +1 t+1 β +1 t t+1 1 +t 1 F +1 t+1 whch, by tang s r + 1 t n 5 n Lea 1.1, gves the claed result. Followng the proof way of Theore.3, we have the followng result: Theore.4. For t > 0, 1 n Ft 5 n 1 5 n +1 F t n L tn [n 0] t F tn f n s even, f n s odd, For odd t > 0, 1 n F +1 t 1 n L n +1t F +1tn, and for even t > 0, 1 n F +1 t 5 n 5 n t F +1tn t L +1tn f n s even f n s odd.

7 Forulas for Bnoal Sus Includng Powers of Fbonacc and Lucas Nubers Bnoal Sus and Bnoal Alternatng Sus for Lucas Nubers Now we consder bnoal and alternatng bnoal sus of powers of Lucas nubers. Theore 3.1. For t > 0, L t 1 L n t L tn + n. For even t > 0, L +1 t + 1 L n +1t L +1tn, and, for odd t > 0, L +1 t 5 n/ t L +1tn 5 n+1/ t F +1tn f n s even, f n s odd. Proof. For t > 0, by the Bnet forula of {L n } and 1, we wrte L t α t + β t 1 n α t + β t + n n L t +, 1 αβ t whch, by tang s t and r 0 n 4 n Lea 1.1, gves the claed result.

8 76 Erah KILIÇ, Iler AKKUS, Neşe ÖMÜR, Yücel Türer ULUTAŞ For t > 0, by the Bnet forula of {L n } and 1, we wrte L +1 t α t + β t α +1 t + β +1 t + 1 n L +1 t whch, by tang s + 1 t and r 0 n 4 n Lea 1.1, gves the claed result. Followng the proof way of the prevous Theores, we gve the followng results wthout proof: Theore 3.. For t > 0, 1 n L +1t For even t > 0, L +1 +1t and, for odd t > 0, L +1 +1t 5 n n Theore 3.3. For t > 0, 1 L t 5 n 1 5 n t L n t L tn t n. L n +1t L +1tn+1, 1 +1t L +1tn t F +1tn+1 t L tn + t F tn f n s even, f n s odd. [n 0] f n s even, f n s odd.

9 Forulas for Bnoal Sus Includng Powers of Fbonacc and Lucas Nubers 77 For odd t > 0, 1 L +1 t 1 n and, for even t > 0, n 5 n 1 L +1 t n+1 Theore 3.4. For t > 0, 1 L +1t 1 5n/ t L +1 tn t F +1 tn n+1/ 1 1 t t L tn+1 1 t [n 0] For odd t > 0, 1 L +1 +1t 1n and, for even t > 0, 1 L +1 +1t 5 n 5 n L n +1 t L +1 tn, 1 t t F tn f n s even, f n s odd. f n s even, f n s odd. 1 L n +1 t L +1 tn+1, +1 t L +1 tn+1 +1 t F +1 tn+1 R E F E R E N C E S f n s even, f n s odd. [1] K. Oze, On Melha s su,the Fbonacc Quarterly, , [] H. Prodnger, On a su of Melha and ts varants, The Fbonacc Quarterly, , [3] S. Vajda, Fbonacc & Lucas nubers, and the golden secton. Theory and applcatons. John Wley & Sons, Inc., New Yor, 1989.

10 78 Erah KILIÇ, Iler AKKUS, Neşe ÖMÜR, Yücel Türer ULUTAŞ [4] M. Weann and C. Cooper, Dvsblty of an F L type convoluton. Applcatons of Fbonacc Nubers, 9 004, [5] E. Kılıç, N. Öür and Y. T. Ulutaş, Alternatng sus of the powers of Fbonacc and Lucas nubers, Msolc Math. Notes, Vol , No. 1, [6] E. Kılıç, N. Öür and Y. T. Ulutaş, Bnoal sus whose coeffcents are products of ters of bnary sequences, Utltas Math., , 45 5.