Continued fractions and irrationality exponents for modified Engel and Pierce series
Andrew N. W. Hone, Juan Luis Varona · Monatshefte für Mathematik · 2018
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.