30 research outputs found
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