Continued fractions and irrationality exponents for modified Engel and Pierce series

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 $j$.