Abstract
The third author discovered numerically an interesting phenomenon that the Aitken acceleration to the ratiop n =f n−1/f n of the successive Fibonacci numbersf n−1 andp n in exactlyp 2n . This fact has a natural extension to the case that Aitken acceleration is replaced by ε-algorithm and Fibonacci numbers can be replaced by solutions of second order difference equations. As a byproduct we will show ann-plication formula of cotz.
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References
M. Arai, K. Okamoto and Y. Kametaka, A new addition formula for cot(x), Aitken-Steffensen acceleration and Cauchy matrix. To appear.
J. H.McCabe and M. Phillips, Fibonacci and Lucas numbers and Aitken and Aitken acceleration. Fibonacci Numbers and Their Applications, D. Reidel, Dordrecht, 1986.
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Arai, M., Okamoto, K. & Kametaka, Y. Aitken acceleration and Fibonacci numbers. Japan J. Appl. Math. 5, 145–152 (1988). https://doi.org/10.1007/BF03167905
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DOI: https://doi.org/10.1007/BF03167905