Discrete Dynamics in Nature and SocietyVolume 2011 (2011), Article ID 284261, 11 pagesdoi:10.1155/2011/284261

Research Article

Some Finite Sums Involving Generalized Fibonacci and Lucas Numbers

E. Kılıç,¹ N. Ömür,² and Y. T. Ulutaş²

¹Mathematics Department, TOBB University of Economics and Technology , Sogutozu, 06560 Ankara, Turkey²Mathematics Department, Kocaeli University, 41380 İzmit Kocaeli, Turkey

Received 11 August 2011; Accepted 7 September 2011

Academic Editor: Gerald Teschl

Copyright © 2011 E. Kılıç et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By considering Melham's sums (Melham, 2004), we compute various more general nonalternating sums, alternating sums, and sums that alternate according to involving the generalized Fibonacci and Lucas numbers.

1. Introduction

Let , , and be assumed to be arbitrary nonzero complex numbers with . Define second-order linear recursion by with for all integers . Since , the roots and of are distinct.

Also define the sequence via the terms of sequence as .

The Binet formulas for the sequences and are where and .

For , , we denote and so , respectively. When , (th Fibonacci number) and (th Lucas number).

Inspired by the well-known identity Clary and Hemenway [1] obtained factored closed-form expressions for all sums of the form , where is an integer. Motivated by the results in [1], Melham [2] computed all sums of the form and . In [3], Melham computed various nonalternating sums, alternating sums, and sums that alternate according to for sequences and . The author gathers his sums in three sets. Here we recall one example from each set for the reader's convenience:

We refer to [4] for general expansion formulas for sums of powers of Fibonacci and Lucas numbers, as considered by Melham, as well as some extensions such that where .

For alternating analogues of the results given by Prodinger, that is, we refer to [5].

Hendel [6] gave the factorization theorem which exhibits factorizations of sums of the form . The author also introduced a unified proof method based on formulae for the factorizations of .

In [7], Curtin et al. derived formulae for the shifted summations and the shifted convolutions for positive integers and arbitrary integers and .

In this paper, our main purpose is to consider Melham's sums involving double products of terms of ,,, and given in [3] and then compute several more general nonalternating sums, alternating sums, and sums that alternate according to .

2. Certain Finite Sums of Double Products of Terms

In this section, we will investigate certain sums consisting of products of at most two terms of : nonalternating sums, alternating sums and sums that alternate according to . From the Binet forms of and , we give the following lemma for further use without proof.

Lemma 2.1. Let , , and be as in Section 1, and let . Then for all integers ,

Theorem 2.2. Fix integers , and . (i)If is even, then for all integers , (i)If is odd, then for all integers ,

Proof. Using the Binet formulas, we compute Since , we can obtain that for even The result follows.

For example, when , and , we obtain

Theorem 2.3. Fix integers , and . Let . (1)If is odd, then equals (2)If is odd and the parities of and are the same, then equals (3)If is odd and the parities of and are the different, then equals

Proof. Consider Since , for odd , we find The result is now obtained by considering the values of .

Theorem 2.4. Fix integers , and . For all integers , where .

Proof. Consider Here we have that and, by , for even . Now formula (2.12) follows. The remaining formulas are proven in a similar manner.

Notice that in (2.12)-(2.13), one limit of summation is even while the other is odd. Accordingly we have observed that each of (2.12)-(2.13) has a dual sum that is obtained with the use of the rule below. We highlight this rule since it also applies to get certain groups of sums in Section 2.

From [3], we recall the rule for the formation of the dual sum.(1)Replace the even limit by the even limit corresponding to the other residue class modulo 4 and the odd limit by the odd limit corresponding to the other residue class modulo 4.(2)Calculate the subscripts on the right in accordance with the paragraph following (2.13).(3)Multiply the right side by .

For example, for odd integer , the dual of (2.13) is where is defined as before.

Theorem 2.5. Fix integers , and . (i)If and have the same parities, then where .(ii)If and have different parities, then where is defined as before and .

Proof. Suppose that and have the same parities. Consider From the definition of , we obtain Since we get Taking in Lemma 2.1, we write Thus the result follows. Similar arguments yield the remaining formulas, where we must consider the parities of .

For example, the dual of (2.17) is given by if and have the same parities, and the dual of (2.18) is given by if and have different parities, where and are defined as before.

Acknowledgment

Thanks are due to the anonymous referee who made suggestions towards a better presentation.

References

S. Clary and P. D. Hemenway, “On sums of cubes of Fibonacci numbers,” in Applications of Fibonacci Numbers, vol. 5, pp. 123–136, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.

R. S. Melham, “Alternating sums of fourth powers of Fibonacci and Lucas numbers,” The Fibonacci Quarterly, vol. 38, no. 3, pp. 254–259, 2000.

R. S. Melham, “Certain classes of finite sums that involve generalized Fibonacci and Lucas numbers,” The Fibonacci Quarterly, vol. 42, no. 1, pp. 47–54, 2004.

H. Prodinger, “On a sum of Melham and its variants,” The Fibonacci Quarterly, vol. 46-47, no. 3, pp. 207–215, 2008.

E. Kılıç, N. Ömür, Y. T. Ulutaş, et al., “Alternating sums of the powers of Fibonacci and Lucas numbers,” Miskolc Mathematical Notes, vol. 12, no. 1, pp. 87–103, 2011.

R. J. Hendel, “Factorizations of ,” The Fibonacci Quarterly, vol. 45, no. 2, pp. 128–132, 2007.

B. Curtin, E. Salter, and D. Stone, “Some formulae for the Fibonacci numbers,” The Fibonacci Quarterly, vol. 45, no. 2, pp. 171–179, 2007.