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Continued fractions for some alternating series. (English) Zbl 0719.11038

We discuss certain simple continued fractions that exhibit a type of “self-similar” structure: their partial quotients are formed by perturbing and shifting the denominators of their convergents. We prove that all such continued fractions represent transcendental numbers. As an application, we prove that Cahen’s constant \(C=\sum_{i\geq 0}\frac{(- 1)^ i}{S_ i-1}\) is transcendental. Here \((S_ n)\) is Sylvester’s sequence, defined by \(S_ 0=2\) and \(S_{n+1}=S^ 2_ n-S_ n+1\) for \(n\geq 0\). We also explicitly compute the continued fraction for the number C; its partial quotients grow doubly exponentially and they are all squares.
Reviewer: J.L.Davison

MSC:

11J70 Continued fractions and generalizations
11J82 Measures of irrationality and of transcendence
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References:

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