Continued fractions for some transcendental numbers

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 $n\ge 1$. 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.

Final example corrected compared with published version. in Monatshefte fur Mathematik, Online First, December 2015

References in corpus (1)

• Curious continued fractions, nonlinear recurrences and transcendental numbers

Cited by in corpus (4)

• Continued fractions and irrationality exponents for modified Engel and Pierce series
• Irrationality exponents of generalized Hone series
• Continued fractions for strong Engel series and Lüroth series with signs
• Transformation formulas of finite sums into continued fractions