Skip to main content
Kent Academic Repository

Continued fractions and irrationality exponents for modified engel and pierce series

Hone, Andrew N.W., Varona, Juan Luis (2018) Continued fractions and irrationality exponents for modified engel and pierce series. Monatshefte fur Mathematik, . pp. 1-16. ISSN 0026-9255. E-ISSN 1436-5081. (doi:10.1007/s00605-018-1244-1) (KAR id:70203)


An Engel series is a sum of reciprocals of a non-decreasing

sequence (xn) 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 ? whose continued fraction

expansion is determined explicitly by the corresponding sequence

(xn), 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 + ?5)/2, and we further

identify infinite families of transcendental numbers ? 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 x2j divides xj+1 for all


Item Type: Article
DOI/Identification number: 10.1007/s00605-018-1244-1
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Andrew Hone
Date Deposited: 20 Nov 2018 11:03 UTC
Last Modified: 09 Jan 2024 11:49 UTC
Resource URI: (The current URI for this page, for reference purposes)

University of Kent Author Information

  • Depositors only (login required):

Total unique views for this document in KAR since July 2020. For more details click on the image.