Continued fractions and irrationality exponents for modified engel and pierce series

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

j.