Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2018
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/1810.10127 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910769773281280 |
|---|---|
| author | Hone, Andrew N. W. Varona, Juan Luis |
| author_facet | Hone, Andrew N. W. Varona, Juan Luis |
| contents | An Engel series is a sum of reciprocals of a non-decreasing sequence $(x_n)$ of positive integers, which is such that each term is divisible by the previous one, and a Pierce series is an alternating sum of the reciprocals of a sequence with the same property. Given an arbitrary rational number, we show that there is a family of Engel series which when added to it produces a transcendental number $α$ whose continued fraction expansion is determined explicitly by the corresponding sequence $(x_n)$, where the latter is generated by a certain nonlinear recurrence of second order. We also present an analogous result for a rational number with a Pierce series added to or subtracted from it. In both situations (a rational number combined with either an Engel or a Pierce series), the irrationality exponent is bounded below by $(3+\sqrt{5})/2$, and we further identify infinite families of transcendental numbers $α$ whose irrationality exponent can be computed precisely. In addition, we construct the continued fraction expansion for an arbitrary rational number added to an Engel series with the stronger property that $x_j^2$ divides $x_{j+1}$ for all $j$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_10127 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Continued fractions and irrationality exponents for modified Engel and Pierce series Hone, Andrew N. W. Varona, Juan Luis Number Theory An Engel series is a sum of reciprocals of a non-decreasing sequence $(x_n)$ of positive integers, which is such that each term is divisible by the previous one, and a Pierce series is an alternating sum of the reciprocals of a sequence with the same property. Given an arbitrary rational number, we show that there is a family of Engel series which when added to it produces a transcendental number $α$ whose continued fraction expansion is determined explicitly by the corresponding sequence $(x_n)$, where the latter is generated by a certain nonlinear recurrence of second order. We also present an analogous result for a rational number with a Pierce series added to or subtracted from it. In both situations (a rational number combined with either an Engel or a Pierce series), the irrationality exponent is bounded below by $(3+\sqrt{5})/2$, and we further identify infinite families of transcendental numbers $α$ whose irrationality exponent can be computed precisely. In addition, we construct the continued fraction expansion for an arbitrary rational number added to an Engel series with the stronger property that $x_j^2$ divides $x_{j+1}$ for all $j$. |
| title | Continued fractions and irrationality exponents for modified Engel and Pierce series |
| topic | Number Theory |
| url | https://arxiv.org/abs/1810.10127 |