Rational function approximations of the special function $e^{x}E_{1}(x)$ and applications to irrationality of Euler-Gompertz constant $\delta$

TL;DR

This paper develops rational function approximations for the special function $e^{x}E_{1}(x)$ using continued fractions, proving convergence and exploring implications for the irrationality of the Euler-Gompertz constant.

Contribution

It introduces a continued fraction method for approximating $e^{x}E_{1}(x)$ and applies it to construct rational sequences related to the Euler-Gompertz constant, offering a potential approach to prove its irrationality.

Findings

Proved convergence of the continued fraction to $F(x)$ for positive $x$

Established inequalities relating approximations to $F(x)$

Constructed rational sequences approaching the Euler-Gompertz constant

Abstract

In \cite{d4}, we gave a method to construct a continued fraction of the function $F(x):=e^{x}E_{1}(x)$. More precisely we define $F_{1}(x)$ as the reciprocal of $F(x)$ and we inductively define $F_{m}(x)$ as the reciprocal of ``$F_{m-1}(x)$ minus the main term of $F_{m-1}(x)$ at infinity''. We calculated the main term of $F_{m}(x)$ at infinity by using \cite[Proposition 2.1]{d4}. This method is analogous to the regular continued fraction expansion of real numbers. \\ \ \ \ \ In this paper we prove that the continued fraction converges to $F(x)$ for any positive real number $x>0$ by following the proof of that the regular continued fraction of a positive and irrational real number $\alpha$ converges to $\alpha$. Essentially we prove inequalities for $Q_{m}(x)$ (in Theorem 4.1) and inequalities $F_{m}(x)>0$ (in Section 5). In particular, we prove stronger inequalities $\displaystyle\frac{P_{2k}(x)}{Q_{2k}(x)}<F(x)<\displaystyle\frac{P_{2k-1}(x)}{Q_{2k-1}(x)}$ (than $F_{m}(x)>0$) and give two proofs of these. In Section 6, we show an asymptotic relation between $Q_{2k}(x)$ and $Q_{2k-1}(x)$ by using properties of the classical Laguerre polynomial. In Section 7, we consider Euler-Gompertz constant $\delta$. As far as we know, irrationality of $\delta$ is still an open problem. We construct a sequence of rationals $\displaystyle\frac{A_{i}}{B_{i}}\ (i=1,2,3,\cdots)$ such that $\delta B_{i}-A_{i}$ approaches 0 as $i$ approaches infinity and give a sufficient condition of that $\delta B_{i}-A_{i}\neq 0$ for any positive integer $i$. Therefore, if it is proved that this condition holds, it completes a proof of irrationality of Euler-Gompertz constant $\delta$.