论文标题
特殊功能的理性函数近似$ e^{x} e_ {1}（x）$和应用于Euler-Gompertz常数$δ$的应用程序
Rational function approximations of the special function $e^{x}E_{1}(x)$ and applications to irrationality of Euler-Gompertz constant $δ$
论文作者
论文摘要
在\ cite {d4}中，我们提供了一种构造函数$ f（x）的持续分数的方法：= e^{x} e_ {1}（x）$。更准确地说，我们将$ f_ {1}（x）$定义为$ f（x）$的倒数，我们将$ f_ {m}（x）$定义为`$ f_ {m-f_ {m-1}（x）$的主要项$ f_ {m-1}（x）的主要项$ f_ f_ {m-1}（m-1}（x）（x）（x）$ at infinity'''''''''的主要项。我们通过使用\ cite [essosing 2.1] {d4}计算了无穷大的$ f_ {m}（x）$的主要项。该方法类似于实数的常规持续分数扩展。 \\\\\\\\ \ \在本文中，我们证明，持续分数收敛到$ f（x）$，对于任何正实际数字$ x> 0 $，遵循证明正常持续的持续分数的正常和不合理的实际数量$α$收敛到$α$。从本质上讲，我们证明了$ q_ {m}（x）$（在定理4.1中）和不等式的不等式$ f_ {m}（x）> 0 $（在第5节中）。特别是，我们证明了$ \ displayStyle \ frac {p_ {2k}（x）} {q_ {2k}（x）} <f（x）<f（x）<\ displayStyle \ frac { $ f_ {m}（x）> 0 $），并给出两个证据。在第6节中，我们通过使用经典的laguerre polyenmial的属性来显示$ q_ {2k}（x）$和$ q_ {2k-1}（x）$之间的渐近关系。在第7节中，我们考虑Euler-Gompertz常数$δ$。据我们所知，$δ$的不合理性仍然是一个空旷的问题。我们构造一系列理性$ \ displayStyle \ frac {a_ {a {i}} {b_ {i}}} \（i = 1,2,3，\ cdots）$，使得$Δb_{i} -a__ {i} -a_ {i} {i}对于任何正整数$ i $，$ΔB_{i} -a_ {i} \ neq 0 $。因此，如果证明这种情况存在，它将完成Euler-Gompertz常数$δ$的非理性证明。
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 $α$ converges to $α$. 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 $δ$. As far as we know, irrationality of $δ$ is still an open problem. We construct a sequence of rationals $\displaystyle\frac{A_{i}}{B_{i}}\ (i=1,2,3,\cdots)$ such that $δB_{i}-A_{i}$ approaches 0 as $i$ approaches infinity and give a sufficient condition of that $δB_{i}-A_{i}\neq 0$ for any positive integer $i$. Therefore, if it is proved that this condition holds, it completes a proof of irrationality of Euler-Gompertz constant $δ$.
