Alternating sums of reciprocal generalized Fibonacci numbers
© Kuhapatanakul; licensee Springer. 2014
Received: 25 July 2014
Accepted: 25 August 2014
Published: 29 August 2014
Recently Holliday and Komatsu extended the results of Ohtsuka and Nakamura on reciprocal sums of Fibonacci numbers to reciprocal sums of generalized Fibonacci numbers. The aim of this work is to give similar results for the alternating sums of reciprocals of the generalized Fibonacci numbers with indices in arithmetic progression. Finally we note our generalizations of some results of Holliday and Komatsu.
AMS Subject Classification
Primary 11B37; secondary 11B39
KeywordsFibonacci numbers Reciprocal sums
If p = 1,2, then U n are called the Fibonacci numbers F n and Pell numbers P n , respectively. Also, if p is any variable x, then U n are called Fibonacci polynomials F n (x).
where ⌊·⌋ is the floor function.
Similar properties were investigated in several different ways; see (Wu and Zhang 2012; Wu and Zhang 2013; Zhang 2011). Recently, (Kuhapatanakul 2013) gave a similar formula (1) for alternating sums of reciprocal generalized Fibonacci numbers.
where a,b are non-negative integers with b < a. We also extend two identities (1) and (2) by replacing U k with Ua k−b.
We begin with some identities of the generalized Fibonacci numbers whose will be used in the proofs of main theorem.
U r U n+1 + U r−1 U n = U n+r.
U r U n−1 − U r−1 U n = (−1)r−1 U n−r.
Every proof is done by induction and omitted.
We are now ready to verify our results.
The numerator of the right hand side of above equation is positive if f(n−1) + n is even. Also, the numerator is negative if f(n−1) + n is odd.
where f(n−1) + n is even.
where f(n−1)+n is odd.
This completes the proof.
If f(n−1) + n = a(n−1) + n − b is even, then we have “both a and b are odd” or “ a,b,n are even” or “a is even and b,n are odd”.
If f(n−1) + n = a(n−1) + n − b is odd, then we have “a is odd and b is even” or “ a,b are even and n is odd” or “ a,n are even and b is odd”.
Now we present some examples of Theorem 1 in the following corollary
Some examples for the Fibonacci numbers.
Sums of reciprocals
We will obtain generalizations of the result of (Holliday and Komatsu 2011) on the reciprocal sums of U k to the reciprocal sums of Ua k−b.
The numerator of the right hand side of above equation is positive if f(n−1) is odd. Also, the numerator is negative if f(n−1) is even.
where a(n−1)−b is odd.
where a(n−1)−b is even.
If f(n−1) = a(n−1)−b is odd, then we have “a is even and b is odd” or “ a,b,n are odd” or “a is odd and b,n are even”.
If f(n−1) = a(n−1)−b is even, then we have “both a and b are even” or “ a,b are odd and n is even” or “ a,n are odd and b is even”.
This research is supported by a research grant for new scholars from the Thailand Research Fund (TRF), Grant No. MRG5680127, and the Kasetsart University Research and Development Institute (KURDI), Thailand.
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