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arXiv · 1509.05019

Continued fractions for some transcendental numbers

Andrew N. W. Hone
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Abstract

We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$. It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number.

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Andrew N. W. Hone. 2016-03-10. Continued fractions for some transcendental numbers. https://doi.org/10.1007/s00605-015-0844-2

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