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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

##### MSC:
 11A55 Continued fractions 11J70 Continued fractions and generalizations 11B37 Recurrences