Continued fractions and irrationality exponents for modified Engel and Pierce series

TL;DR

The paper addresses the problem of representing sums of Engel and Pierce series appended to a rational number as explicit continued fractions, and it establishes transcendence and precise Diophantine properties of the resulting numbers. It develops a nonlinear second-order recurrence generating the index sequence $(x_n)$, and derives explicit CF expansions for $\alpha=\frac{p}{q}+\sum_{j=2}^\infty \frac{1}{x_j}$ (Engel) and $\alpha=\frac{p}{q}\pm\sum_{j=2}^\infty \frac{(-1)^j}{x_j}$ (Pierce), including generalizations with a parameter $m$. The authors prove a universal lower bound on the irrationality exponent, $\mu(\alpha)\ge\frac{3+\sqrt{5}}{2}$, and show how to realize exact values $\mu(\alpha)=\lambda$ for $\lambda=(d+2+\sqrt{d(d+4)})/2$ (via polynomially growing $u_n$) or any $\nu\ge\mu^*=(3+\sqrt{5})/2$ (via controlled exponential growth), thereby producing broad families of transcendental numbers with computable $\mu(\alpha)$. Additionally, the paper extends these constructions to Engel series with the stronger divisibility $x_j^2\mid x_{j+1}$, giving explicit CFs through CF-transformations and concrete examples. These results advance explicit transcendence proofs and quantitative Diophantine properties for Engel/Pierce-type expansions beyond previous work of Hone and others.

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 $α$ 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 $α$ 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 $j$.

Theorems & Definitions (13)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Lemma 3.1
• Corollary 3.2
• Theorem 3.3
• Theorem 3.4
• Lemma 4.1