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Aitken acceleration and Fibonacci numbers. (English) Zbl 0649.10008

Let \(p_ n=g_ n/f_ n\) where \(f_ n\) and \(g_ n\) are the solutions of the same second order difference equation with the initial data \(f_ 0=0\), \(f_ 1=1\) and \(g_ 0\neq 0\). Then it is proved that the m-th Aitken acceleration to \(p_ n\) is exactly \(p_{mn}\). A continuous version is also presented.
Reviewer: R.F.Tichy

MSC:

11B37 Recurrences
65B05 Extrapolation to the limit, deferred corrections

Citations:

Zbl 0589.10011
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References:

[1] M. Arai, K. Okamoto and Y. Kametaka, A new addition formula for cot(x), Aitken-Steffensen acceleration and Cauchy matrix. To appear. · Zbl 0589.10011
[2] J. H.McCabe and M. Phillips, Fibonacci and Lucas numbers and Aitken and Aitken acceleration. Fibonacci Numbers and Their Applications, D. Reidel, Dordrecht, 1986. · Zbl 0589.10010
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