Selçuk J. Appl. Math. Selçuk Journal of Special Issue. pp. 47-50, 2012 Applied Mathematics
On the Sums of Powers of -Fibonacci and -Lucas Sequences Yasin Yazlik, Nazmiye Yilmaz, Necati Taskara
Selcuk University, Science Faculty Department of Mathematics, 42250, Konya, Turkiye e-mail: yyazlik@ selcuk.edu.tr,nzyilm az@ selcuk.edu.tr,ntaskara@selcuk.edu.tr
Presented in 3 National Communication Days of Konya Eregli Kemal Akman Vocational School, 28-29 April 2011.
Abstract. In this study, we investigate some proporties additive of -Fibonacci and -Lucas sequences and obtain new identities on sums of powers these se-quences.
Key words: -Fibonacci, -Lucas numbers.
2000 Mathematics Subject Classification: 11B39, 11C20. 1. Introduction and Preliminaries
The Fibonacci numbers are defined for all ≥ 1 by the recurrence relation +1= +−1, where 0= 0 and 1= 1. The Lucas numbers are defined for all ≥ 1 by the same recurrence relation, where 0= 2 and 1= 1. There are so many studies in the literature that concern about Fibonacci and Lucas sequences in [3-6]. For instance, Clary and Hemenway [3] discovered factored closed-form expressions for all sums of the form
X =1
3
where r is an integer. In [4], Jennings presented some remarkable elementary identities for sums of powers of reciprocals of Fibonacci and Lucas numbers. In [5], Melham proved a theorem involving a sum of products of Fibonacci numbers and corresponding theorem for Lucas numbers. In [6], Alex and Hongwei obtained formula which gave powers of Fibonacci numbers by using the generating functions.
We should recall that, for ∈ R+ in [1,2], it has been defined -Fibonacci {}∈Nand -Lucas {}∈Nsequences by the recursive equations (1) +2= +1+ and +2= +1+
with initial conditions 0= 0 1= 1 and 0= 2, 1 = respectively. Obviously, if we take = 1 in (1), then these two sequences reduce to the well-known Fibonacci and Lucas sequences, respectively. Thus it can be clearly obtained the characteristic equation of (1) as the form 2= + 1 having the roots
= + √ 2+ 4 2 and = −√2+ 4 2
It means that the following relations hold for the numbers , :
(2) + = − =p2+ 4 = −1 Hence the Binet formulas
(3) =
−
− and = + can be thought as solutions of the recursive equations in (1).
In this study, we obtain the recurrence relations for powers of -Fibonacci and -Lucas sequences.
2. Main Results
Let us first consider the following lemmas which can be proved easily by iteration process. In fact, by these lemmas, they will be given new formulas for the powers of -Fibonacci and -Lucas sequences.
Lemma 1. For ≥ 0 we have i) 2 +3= ¡ 2+ 1¢2 +2+ ¡ 2+ 1¢2 +1− 2 ii) 2+3=¡2+ 1¢2+2+¡2+ 1¢2+1− 2
Proof. In here, we will prove (i) since (ii) can be thought in the same manner with it.
i) For ≥ 0 , we know +3 = +2+ +1 from (1). If ıt is multiplying this equation with +3, then we have
+32 = +2+3+ +1+3
= 2+22 + (2+ 1)+12 + +1+ +2+1
= 2+22 + (2+ 1)+12 + (+2− ) + +2+1 = ¡2+ 1¢+22 +¡2+ 1¢+12 − 2
from where the result is obtained. Lemma 2. For ≥ 0 we get
i) 3 +4= ¡ 3+ 2¢3 +3+ ¡ 4+ 22+ 2¢3 +2− ¡ 3+ 2¢3 +1− 3 ii) 3+4=¡3+ 2¢3+3+¡4+ 22+ 2¢3+2−¡3+ 2¢3+1− 3
Proof. As contrary, in here we will just prove (ii) since the proof of (i) can be done quite similarly with it.
ii)Let be =¡3+ 2¢3+3+¡4+ 22+ 2¢3+2−¡3+ 2¢3+1− 3
By replacing Binet Formula for the -Lucas numbers in (3) ¡
3+ 2¢3+3+¡4+ 22+ 2¢3+2−¡3+ 2¢3+1− 3 and taking into account = + in (2), we find out
= ¡3+ 3− − ¢ ¡+3+ +3¢3+¡4+ 4− 2− 2+ 2¢ ¡+2+ +2¢3 −¡3+ 3− − ¢ ¡+1+ +1¢3− (+ )3
If we make the arrangements and take into account = −1 in (2), then it is obtained the equality
= 3+12+ 3(−1)+4+4+ 3(−1)+4+4+ 3+12 = ¡+4+ +4¢3
= 3+4 which ends up the proof. Lemma 3. For ≥ 0 we get
i) 4 +5= ¡ 4+ 32+ 1¢4 +4+ ¡ 6+ 54+ 72+ 2¢4 +3− −¡6+ 54+ 72+ 2¢+24 −¡4+ 32+ 1¢+14 + 4 ii) 4 +5= ¡ 4+ 32+ 1¢4 +4+ ¡ 6+ 54+ 72+ 2¢4 +3− −¡6+ 54+ 72+ 2¢4+2−¡4+ 32+ 1¢4+1+ 4
Proof. Proof of this Lemma can be seen easily in a similar manner with Lemma 2.
The following Theorem gives us the recurrence relations for powers of -Fibonacci and -Lucas sequences.
Theorem 1. For ≥ 0 and we have i) = +1 X =1 (−1)+(+1)(+2)2 − where = ⎧ ⎪ ⎨ ⎪ ⎩ +1 = 1 Y =1 +2− 6= 1 ii) = +1 X =1 (−1)+(+1)(+2)2 − where = ⎧ ⎪ ⎨ ⎪ ⎩ +1 = 1 Y =1 +2− 6= 1
Proof. In here, we will prove (i) since (ii) can be thought in the same manner with it.
i) If we consider the equation (1) then it can be clearly written as +2= 2+1+
21 21
If we apply same idea to Lemma 1, then +32 = ¡2+ 1¢+22 +¡2+ 1¢+12 − 2 = 3+22 + 32 21 +12 −321 321 If we apply same idea to Lemma 2, then
+43 = ¡3+ 2¢+33 +¡4+ 22+ 2¢+23 −¡3+ 2¢+13 − 3 = 4+33 + 43 21 +23 −432 321 +13 −4321 4321 3 By keeping the this sequence and using same tecnique, we get
5+44 + 54 21 +34 −543 321 +24 −5432 4321 +14 +54321 54321 4 which gives 4
+5in the statement of Lemma 3. So, by iterating this above all progresses, we obtain the general term as the form of
+1 X =1 (−1)+(+1)(+2)2 − = ⎧ ⎪ ⎨ ⎪ ⎩ +1 = 1 Y =1 +2− 6= 1 that implies as required. References
1. Falcon S., Plaza A ., "On the Fibonacci-numbers", Chaos, Solitons & Fractals, 32(5):1615—24 (2007).
2. Falcon S., "On the-Lucas Numbers", Int. J. Contemp. Math. Sciences 6(21), 1039-1050 (2011).
3. Clarly S. & Hemenway P:D., "On sums of cubes of Fibonacci Numbers", In Appli-cations of Fibonacci Numbers 5,123-136 (1993).
4. Jennings D., "On sums the reciprocals of Fibonacci and Lucas Numbers", The Fibonacci Quarterly, 32(1), 18-21, (1994).
5. Melham R.S., "Sums of certain products of Fbonacci and Lucas Numbers", The Fibonacci Quarterly 37(3), 248-251 (1999).
6. Chen A. and Chen H., "Identities for the Fibonacci Powers", International Journal of Mathemtical Education 39(4), 534-541, (2008).
