Closed forms for certain fibonacci type sums that involve second order products

In this paper, we present closed forms for certain finite sums in which the summand is a product of generalized Fibonacci numbers. We present our results in the form of six theorems that feature a generalized Fibonacci sequence {Wn}, and an accompanying sequence {Wn}- We add a further layer of generalization to our results with the use of three parameters s, k, and m. The inspiration for this paper comes from a website of Knott that lists so-called order 2 summations involving the Fibonacci and Lucas numbers. Probably the most well-known of these summations is σni=1Fi2=FnFn+1

Finite sums that involve reciprocals of products of generalized Fibonacci numbers

© 2014 Walter de Gruyter GmbH, Berlin/Boston. In this paper we find closed forms for certain finite sums. In each case the denominator of the summand consists of products of generalized Fibonacci numbers. Furthermore, we express each closed form in terms of rational numbers

Sums of certain products of fibonacci & Lucas numbers-part III

For the Fibonacci numbers, the summation formula σnk=1 Fk2=FnFn+1is well-known. Its charm lies in the fact that the right side is a product of terms from the Fibonacci sequence. In the earlier paper [5], the author presents similar formulas where, in each case, the right side consists of arbitrarily long products of an even number of distinct terms from the Fibonacci sequence. The formulas in question contain several parameters, and this contributes to their generality. In this paper, we provide additional results of a similar nature where the right side consists of arbitrarily long products of an odd number of distinct terms from the Fibonacci sequence. Most of the results that we present apply to a sequence that generalizes both the Fibonacci and Lucas numbers

Closed forms for finite sums of weighted products of generalized Fibonacci numbers

In this paper, we present closed forms for certain finite sums of weighted products of generalized Fibonacci numbers. Indeed, we present seven multi-parameter families of such finite sums, all of which we believe to be new. In each of these families, the number of factors in the summand is governed by the size of the integer parameter j ≥ 1, and can be made as large as we please. We present our main results in terms of sequences that generalize the Fibonacci/Lucas numbers. Consequently, each of our main results can be specialized to involve the Fibonacci/Lucas numbers. For instance, as a consequence of one of our main results, it follows that (Equation presented) Here the weight term in the summand is 2i-1