On Reciprocal Series of Generalized Fibonacci Numbers with Subscripts in Arithmetic Progression
We investigate formulas for closely related series of the forms: , , for certain values of , , and .
Let be a nonzero integer such that . The generalized Fibonacci and Lucas sequences are defined by the following recurrences: where , and ,, respectively. When , (th Fibonacci number) and (th Lucas number).
If and are the roots of equation , the Binet formulas of the sequences and have the forms: respectively.
In , Backstrom developed formulas for closely related series of the form: for certain values of , , and . For example, he obtained the following series: where represents an odd integer and is an integer in the range to inclusive. Also, he gave the similar results for Lucas numbers.
In , Popov found in explicit form series of the form: for certain values of , , , and .
In , Gauthier found the closed form expressions for the following sums: where for an indeterminate, the generalized Fibonacci and Lucas polynomials and are given by the following recurrences: respectively.
In this paper, we investigate formulas for closely related series of the forms: for certain values of , and .
2. On Some Series of Reciprocals of Generalized Fibonacci Numbers
In this section, firstly, we will give the following lemmas for further use.
Lemma 2.1. Let be an arbitrary nonzero integer. For integer , and for integer ,
Lemma 2.2. For arbitrary integers and ,
Proof. From Binet formulas of sequences and , the desired results are obtained.
Theorem 2.3. For an odd integer ,
Proof. By replacing with in (2.5), we have
Taking and in the equality , the equality (2.8) is rewritten as follows:
We have the sum
For an odd integer , we have
and taking and in identity :
Substituting (2.11) and (2.13) in (2.10), we have the desired result.
For example, if we take and in (2.6), we have Note that
Corollary 2.4. For an odd integer ,
Proof. Using the equalities and in Theorem 2.3, the results are obtained.
Corollary 2.5. Let be an odd integer. For and , and for and ,
Proof. Since the results are easily seen by equalities (2.16).
Theorem 2.6. For an integer and an arbitrary nonzero integer ,
Corollary 2.7. For an arbitrary nonzero integer ,
Theorem 2.8. For an integer and an arbitrary nonzero integer ,
Proof. The proof of the theorem is similar to the proof of Theorem 2.6.
Corollary 2.9. For an arbitrary nonzero integer ,
For example, if we take and in (2.27), we have
Theorem 2.10. For an integer and an arbitrary nonzero integer ,
Proof. The proof of theorem is similar to the proof of Theorem 2.6.
Corollary 2.11. For an arbitrary nonzero integer ,
For example, if we take in the equality (2.30), we have
N. Gauthier, “Solution to problem H-680,” The Fibonacci Quarterly, vol. 49, no. 1, pp. 90–92, 2011.View at: Google Scholar
S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications, John Wiley & Sons, New York, NY, USA, 1989.