Continued fractions and irrationality exponents for modified Engel and Pierce series

TL;DR

This paper constructs transcendental numbers from rational numbers combined with Engel or Pierce series, explicitly determines their continued fractions, and establishes bounds and exact values for their irrationality exponents.

Contribution

It introduces new methods to generate transcendental numbers with explicitly known continued fractions and irrationality exponents from rational numbers and special series.

Findings

Irrationality exponent bounded below by (3+√5)/2.

Explicit continued fraction expansions for constructed transcendental numbers.

Infinite families with computable irrationality exponents.

Abstract

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 $\alpha$ 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 $\alpha$ 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 $j$.