Continued fractions for strong Engel series and Lüroth series with signs

Andrew N. W. Hone ¹   ; Juan Luis Varona ²

¹ School of Mathematics, Statistics and Actuarial Science University of Kent Canterbury CT2 7FS, UK
² Departamento de Matemáticas y Computación Universidad de La Rioja 26006 Logroño, Spain

Acta Arithmetica, Tome 199 (2021) no. 1, pp. 55-75

Résumé

An Engel series is a sum of reciprocals $\sum _{j\geq 1} 1/x_j$ of a non-decreasing sequence of positive integers $x_n$ with the property that $x_n$ divides $x_{n+1}$ for all $n\geq 1$. In previous work, we have shown that for any Engel series with the stronger property that $x_n^2$ divides $x_{n+1}$, the continued fraction expansion of the sum is determined explicitly in terms of $z_1=x_1$ and the ratios $z_n=x_n/x_{n-1}^2$ for $n\geq 2$. Here we show that when this stronger property holds, the same is true for a sum $\sum _{j\geq 1}\epsilon _j/x_j$ with an arbitrary sequence of signs $\epsilon _j=\pm 1$. As an application, we provide explicit continued fractions for particular families of Lüroth series and alternating Lüroth series defined by non-linear recurrences of second order. We also calculate exact irrationality exponents for certain families of transcendental numbers defined by such series.

Détail

DOI : 10.4064/aa200529-26-11

Keywords: engel series sum reciprocals sum geq non decreasing sequence positive integers property divides geq previous work have shown engel series stronger property divides continued fraction expansion sum determined explicitly terms ratios n n geq here stronger property holds sum sum geq epsilon arbitrary sequence signs epsilon application provide explicit continued fractions particular families roth series alternating roth series defined non linear recurrences second order calculate exact irrationality exponents certain families transcendental numbers defined series