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Explicit formulae for Cantor series representing quadratic irrationals. (English) Zbl 0608.10013
Number theory and combinatorics, Proc. Conf., Tokyo/Jap., Okayama/Jap. and Kyoto/Jap. 1984, 369-381 (1985).
[For the entire collection see Zbl 0601.00003.]
W. Sierpiński [Elementary theory of numbers (1964; Zbl 0122.044), p. 279] proved \(\Theta (x)=(1/2)(x-\sqrt{x^ 2- 4})=\sum^{\infty}_{k=1}(b_ 1...b_ k)^{-1}\) for integers \(x>2\), where \(b_ 1=x\), \(b_{k+1}=b^ 2_ k-2\) for \(k\geq 1\). The author considers the continued fraction expansion of the quadratic irrational units \(\Theta\) (x). If \(p_ n/q_ n\) denotes the n-th approximant in this expansion then it is shown that \(\sum^{n}_{k=1}(b_ 1...b_ k)^{-1}=p_{2^{n+1}-2}/q_{2^{n+1}-2}.\) Sierpinski’s result is generalized to a certain class of quadratic irrationals which is obtained from a recurrence relation involving four (instead of one, as with the \(b_ k)\) rational parameters.
Reviewer: G.Köhler

11A55 Continued fractions
11J70 Continued fractions and generalizations
11B37 Recurrences