The Scientific World Journal

Volume 2014 (2014), Article ID 521358, 6 pages

http://dx.doi.org/10.1155/2014/521358

Research Article

On the Higher Power Sums of Reciprocal Higher-Order Sequences

Zhengang Wu¹ and Jin Zhang²

¹Department of Mathematics, Northwest University, Xi'an, Shaanxi, China

²School of Mathematics and Computer Engineering, Xi'an University of Arts and Science, Xi'an, Shaanxi, China

Received 21 November 2013; Accepted 29 January 2014; Published 10 March 2014

Academic Editors: A. Fiorenza and E. Kılıç

Copyright © 2014 Zhengang Wu and Jin Zhang. 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.

Let be a higher-order linear recursive sequence. In this paper, we use the properties of error estimation and the analytic method to study the reciprocal sums of higher power of higher-order sequences. Then we establish several new and interesting identities relating to the infinite and finite sums.

1. Introduction

The so-called Fibonacci zeta function and Lucas zeta function, defined by

where the and denote the Fibonacci numbers and Lucas numbers, have been considered in several different ways; see [1, 2]. Ohtsuka and Nakamura [3] studied the partial infinite sums of reciprocal Fibonacci numbers and proved the following conclusions:

where denotes the floor function.

Further, Wu and Zhang [4, 5] generalized these identities to the Fibonacci polynomials and Lucas polynomials. Various properties of the Fibonacci polynomials and Lucas polynomials have been studied by many authors; see [6–13].

Recently, some authors considered the nearest integer of the sums of reciprocal Fibonacci numbers and other well-known sequences and obtained several meaningful results; see [14–16]. In particular, in [16], Kılıç and Arıkan studied a problem which is a little different from that of [3], namely, that of determining the nearest integer to . Specifically, suppose that (the nearest integer function) and is an integer sequence satisfying the recurrence formula

for any positive integer and . Then we can conclude that there exists a positive integer such that

for all .

In [17], Wu and Zhang unified the above results by proving the following conclusion that includes all the results, [3–8, 15, 16], as special cases.

Proposition 1. For any positive integer , the th-order linear recursive sequence is defined as follows:

with initial values for and at least one of them not being zero. For any positive real number and any positive integer , there exists a positive integer such that

In particular, taking , there exists a positive integer such that

It seems difficult to deal with for all integers , because it is quite unclear a priori what the shape of the result might be. In [18], Xu and Wang applied the method of undetermined coefficients and constructed a number of delicate inequalities in order to study the infinite sum of the cubes of reciprocal Pell numbers and then obtained the following meaningful result.

Proposition 2. For any positive integer , we have the identity

To find and prove this result is a substantial achievement since such a complex formula would not be clear beforehand that a result would even be possible. However, there is no research considering the higher power () of reciprocal sums of some recursive sequences. The main purpose of this paper is using the properties of error estimation and the analytic method to study the higher power of the reciprocal sums of and obtain several new and interesting identities. The results are as follows.

Theorem 3. Let be an th-order sequence defined by (5) with the restrictions and . For any real number and positive integer , where are the roots of the characteristic equation of and , then there exists a positive integer such that

Taking , from Theorem 3 we may immediately deduce the following.

Corollary 4. Let be an th-order sequence defined by (5) with the restrictions and . For positive integer , where are the roots of the characteristic equation of and , then there exists a positive integer such that

For positive real number , whether there exits an identity for

is an interesting open problem.

2. Several Lemmas

To complete the proof of our theorem, we need two lemmas.

Lemma 5. Let with and with . Then for the polynomial

we have the following:

(I)polynomial has exactly one positive real zero with ;(II)other zeros of lie within the unit circle in the complex plane.

Proof. See Lemma 1 of [16].

Lemma 6. Let and be an integer sequence satisfying the recurrence formula (5). Then for any positive integer , we have

where , , and is the positive real zero of .

Proof. From Lemma 2 of [16], the closed formula of is given by

where , , , and is the positive real zero of . Now we prove Lemma 6 by mathematical induction. From formula (14), we have

That is, the lemma holds for . Suppose that for all integers we have

Then for we have

That is, Lemma 6 also holds for . This completes the proof of Lemma 6 by mathematical induction.

3. Proof of Theorem 3

In this section, we shall complete the proof of Theorem 3. From the geometric series as , we have

Using Lemma 6, we have

Consequently,

where .

Taking the reciprocal of this expression yields

Case 1. If , then for any real number and positive integer we have

Case 2. If , for any positive integer , holds. Then for any positive integer with

we have

In both cases, it follows that for any real number and positive integer there exists sufficiently large so that the modulus of the last error term of identity (21) becomes less than . This completes the proof of Theorem 3.

Proof of Corollary 4. From identity (19), we have

Consequently,

Taking the reciprocal of this expression yields

For any positive integer with

we have

So there exists sufficiently large so that the modulus of the last error term of identity (27) becomes less than . This completes the proof of Corollary 4.

4. Computation

We can determine the power of different sequence by MATHEMATICA as the following examples.

Example 7. Let be the second-order linear recursive sequence (see Table 1).

Table 1

2.4142
2.4142
9.3653

3.3028
3.3028
24.8500

4.2361
4.2361
50.3460

5.1926
5.1926
87.1630

Example 8. Let be the third-order linear recursive sequence (see Table 2).

Table 2

2.5468
1.5959
5.8020

3.6274
1.9044
10.1772

4.7246
1.5370
11.9086

5.8074
1.2050
12.9970

Example 9. Let be the fifth-order linear recursive sequence (see Table 3).

Table 3

Therefore, we may immediately deduce the following corollaries.

Corollary 10. Let be the Pell numbers. For any real number and positive integer , there exists a positive integer such that

Corollary 11. Let be the generalized Tribonacci numbers. For any real number and positive integer , there exists a positive integer such that

Corollary 12. Let be a fifth-order sequence. For any real number and positive integer , there exists a positive integer such that

5. Related Results

The following results are obtained similarly.

Theorem 13. Let be an th-order sequence defined by (5) with the restrictions and . Let and be positive integers with . For any real number and positive integer , where are the roots of the characteristic equation of and , then there exist positive integers , , and depending on , and such that the following hold.

(a). (b). (c).

For , we deduce the following identity of infinite sum as special case of Theorem 13.

Corollary 14. Let be an th-order sequence defined by (5) with the restrictions and . Let and be positive integers with . For any positive integer , where are the roots of the characteristic equation of and , then there exist positive integers , , and depending on , , and such that the following hold.

(d). (e). (f).

Proof. We shall prove only (c) in Theorem 13 and other identities are proved similarly. From identity (19), we have

Consequently,

where .

Taking the reciprocal of this expression yields

Case  1. If , then for any real number and positive integer we have

Case  2. If , for any positive integer , holds. Then for any positive integer with

we have

In both cases, it follows that for any real number and positive integer , there exists sufficiently large so that the modulus of the last error term of identity (35) becomes less than . This completes the proof of Theorem 13(c).

Conflict of Interests

The authors declare that there is no conflict interests regarding the publication of this paper.

Acknowledgments

The authors express their gratitude to the referee for very helpful and detailed comments. This work is supported by the N.S.F. (11371291), the S.R.F.D.P. (20136101110014), the N.S.F. (2013JZ001) of Shaanxi Province, and the G.I.C.F. (YZZ12062) of NWU, China.

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