On the reciprocal sums of higher-order sequences
© Wu and Zhang; licensee Springer 2013
Received: 22 May 2013
Accepted: 4 June 2013
Published: 27 June 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.
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.
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].
where denotes the nearest integer. (Clearly, .)
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.
Taking , from Theorem 1 we may immediately deduce the following.
is an interesting open problem.
2 Several lemmas
To complete the proof of our theorem, we need the following.
Polynomial has exactly one positive real zero α with .
Other zeros of lie within the unit circle in the complex plane.
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.
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 .
This contradicts with (2). □
Claim 2 has no complex zero with .
So, we have , which contradicts with (5). □
Claim 3 On the circle and , has the unique zero α.
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. □
where , , and is the positive real zero of .
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 .
which completes the proof (note that if all the roots of are distinct, we can choose and ). □
3 Proof of Theorem 1
Since , there exists sufficient large so that the modulus of the last error term becomes less than , which completes the proof.
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.
For , we deduce the following identity of infinite sum as a special case of Theorem 2.
Since , there exists sufficiently large so that the modulus of the last error term becomes less than , which completes the proof. □
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.
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