- Open Access
On the reciprocal sums of higher-order sequences
Advances in Difference Equations volume 2013, Article number: 189 (2013)
Let be a higher-order recursive sequence. Several identities are obtained for the infinite sums and finite sums of the reciprocals of higher-order recursive sequences.
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. Navas  discussed the analytic continuation of these series. Elsner et al.  obtained that for any positive distinct integer , , , the numbers , , and are algebraically independent if and only if at least one of , , is even.
Ohtsuka and Nakamura  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. Similar properties were also investigated in [6–8]. Related properties of the Fibonacci polynomials and Lucas polynomials can be found in [9–12].
Recently, some authors considered the nearest integer of the sums of reciprocal Fibonacci numbers and other famous sequences and obtained several new interesting identities, see  and . Kilic and Arikan  defined a k th-order linear recursive sequence for any positive integer and as follows:
and they proved that there exists a positive integer such that
where denotes the nearest integer. (Clearly, .)
In this paper, we unify the above results by proving some theorems that include all the results, [3–8] and [13–15], as special cases. We consider the following type of higher-order recurrence sequences. For any positive integer , we define m th-order linear recursive sequences for as follows:
with initial values for and at least one of them is not zero. If , , then are the Fibonacci numbers. If , , , then are the Pell numbers. Our main results are the following.
Theorem 1 Let be an mth-order sequence defined by (1) with the restriction . For any positive real number , there exists a positive integer such that
Taking , from Theorem 1 we may immediately deduce the following.
Corollary 1 Let be an mth-order sequence defined by (1) with the restriction . Then there exists a positive integer such that
For a 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 the following.
Lemma 1 Let be positive integers with and with . Then, for the polynomial
Polynomial has exactly one positive real zero α with .
Other zeros of lie within the unit circle in the complex plane.
Proof For any positive integer and , we have
Thus there exits a positive real zero α of with . According to Descarte’s rule of signs, has at most one positive real root. So, has exactly one positive real zero α with . This completes the proof of (I) in Lemma 1.
Observe that from (I) in Lemma 1 we have
Since has exactly one positive real zero α, has two positive real zeros α and 1. Observe that
To complete the proof of (II) in Lemma 1, it is sufficient to show that there is no zero on and outside of the unit circle. □
Claim 1 has no complex zero with .
Proof Assume that there exits such a zero. So, we have
then we obtain
This contradicts with (2). □
Claim 2 has no complex zero with .
Proof Assume that there exits such a zero. Since ,
then we obtain
So, we have , which contradicts with (5). □
Claim 3 On the circle and , has the unique zero α.
Proof If , then
then we obtain
If or , then , so (6) must be an equality. Therefore, and all lie on the same ray issuing from the origin. Since , are all the elements of , must be the elements of . Therefore we obtain . On the circle and , there are two conditions or . Since , α is the unique zero of , Claim 3 holds.
From the three claims, (II) in Lemma 1 is proven. □
Lemma 2 Let and let be an integer sequence satisfying the recurrence formula (1). Then the closed formula of is given by
where , , and is the positive real zero of .
Proof Let be the distinct roots of , where is the characteristic equation of the recurrence formula (1). From Lemma 1 we know that α is the simple root of , then let , for , denote the multiplicity of the root . From the properties of m th-order linear recursive sequences, can be expressed as follows:
For example, for positive integers , if is the simple root of , then , where , and ; if is the double root of , then , where , and ; if is the multiple root of with the multiplicity , then , where , and .
From Lemma 1 we have for . Since each term of tail in (7) goes to 0 as , we can find the constant and with for such that
which completes the proof (note that if all the roots of are distinct, we can choose and ). □
3 Proof of Theorem 1
In this section, we shall complete the proof of Theorem 1. From the geometric series as , we have
Using Lemma 2, we have
Taking reciprocal, we get
Since , there exists sufficient large so that the modulus of the last error term becomes less than , which completes the proof.
Proof of Corollary 1 From identity (8), we have
Taking reciprocal, we get
So, there exists sufficiently large so that the modulus of the last error term becomes less than , which completes the proof. □
4 Related results
The following results are obtained similarly.
Theorem 2 Let be an mth-order sequence defined by (1) with the restriction . Let p and q be positive integers with . For any real number , there exist positive integers , and depending on , and such that
For , we deduce the following identity of infinite sum as a special case of Theorem 2.
Corollary 2 Let be an mth-order sequence defined by (1) with the restriction . Let p and q be positive integers with . Then there exist positive integers , and depending on , and such that
Proof We shall prove only (c) in Theorem 2 and other identities are proved similarly. From Lemma 2 we have
Taking reciprocal, we get
Since , there exists sufficiently large so that the modulus of the last error term becomes less than , which completes the proof. □
Navas L: Analytic continuation of the Fibonacci Dirichlet series. Fibonacci Q. 2001, 39: 409-418.
Elsner C, Shimomura S, Shiokawa I: Algebraic relations for reciprocal sums of odd sums of Fibonacci numbers. Acta Arith. 2011, 148(3):205-223. 10.4064/aa148-3-1
Ohtsuka H, Nakamura S: On the sum of reciprocal Fibonacci numbers. Fibonacci Q. 2008/2009, 46/47: 153-159.
Wu Z, Zhang W: The sums of the reciprocal of Fibonacci polynomials and Lucas polynomials. J. Inequal. Appl. 2012, 134: 1-8.
Wu Z, Zhang W: Several identities involving Fibonacci polynomials and Lucas polynomials. J. Inequal. Appl. 2013, 205: 1-14.
Holliday S, Komatsu T: On the sum of reciprocal generalized Fibonacci numbers. Integers 2011, 11: 441-455.
Zhang G: The infinite sum of reciprocal of the Fibonacci numbers. J. Math. Res. Expo. 2011, 31: 1030-1034.
Zhang W, Wang T: The infinite sum of reciprocal Pell numbers. Appl. Math. Comput. 2012, 218: 6164-6167. 10.1016/j.amc.2011.11.090
Falcón S, Plaza Á: On the Fibonacci k-numbers. Chaos Solitons Fractals 2007, 32: 1615-1624. 10.1016/j.chaos.2006.09.022
Ma R, Zhang W: Several identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Q. 2007, 45: 164-170.
Wang T, Zhang W: Some identities involving Fibonacci, Lucas polynomials and their applications. Bull. Math. Soc. Sci. Math. Roum. 2012, 55: 95-103.
Yi Y, Zhang W: Some identities involving the Fibonacci polynomials. Fibonacci Q. 2002, 40: 314-318.
Komatsu T: On the nearest integer of the sum of reciprocal Fibonacci numbers. Aportaciones Matematicas Investigacion 2011, 20: 171-184.
Komatsu T, Laohakosol V: On the sum of reciprocals of numbers satisfying a recurrence relation of order s. J. Integer Seq. 2010., 13: Article ID 10.5.8
Kilic E, Arikan T: More on the infinite sum of reciprocal usual Fibonacci, Pell and higher order recurrences. Appl. Math. Comput. 2013, 219: 7783-7788. 10.1016/j.amc.2013.02.003
The authors express their gratitude to the referee for very helpful and detailed comments. This work is supported by the N.S.F. (11071194, 11001218) of P.R. China and G.I.C.F. (YZZ12062) of NWU.
The authors declare that they have no competing interests.
ZW obtained the theorems and completed the proof. HZ corrected and improved the final version. Both authors read and approved the final manuscript.
About this article
Cite this article
Wu, Z., Zhang, H. On the reciprocal sums of higher-order sequences. Adv Differ Equ 2013, 189 (2013). https://doi.org/10.1186/1687-1847-2013-189
- infinite sums
- finite sums
- higher-order recurrences