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## 翻译论文001：Wallis不等式的最佳界

Wallis不等式的最佳界

CHAO-PING   CHEN   AND   FENG   QI

(Communicated by Carmen C. Chicone)

1.介绍

$\text{(2)}$中$P_n$的上下界频繁被数学家引用和应用。$\text{(2)}$中最小上界$\frac{2}{\sqrt{(4n+1)\pi}}$和最大下界$\frac{\sqrt{2}}{\sqrt{(2n+1)\pi}}$，即是，不等式$\frac{{\sqrt 2 }}{{\sqrt {\left( {2n + 1} \right)\pi } }} < {P_n} < \frac{2}{{\sqrt {\left( {4n + 1} \right)\pi } }}\tag{3}$是由N. D. Kazarino得到的 。见【16，pp.47-48和pp.65-67】。我们可以把不等式$\text{(3)}$改写成$\frac{1}{{\sqrt {\pi \left( {n + \frac{1}{2}} \right)} }} < {P_n} < \frac{2}{{\sqrt {\pi \left( {n + \frac{1}{4}} \right)} }}\tag{4}$，对$n\in \mathbb{N}$均成立。

Wallis公式的使用Hadamard乘积【10】从Riemann zeta 函数$\zeta(s)$，由$\zeta '\left( 0 \right)$得到的一个归功于Y. L. Yung的推导可以在【12】中被发现。Wallis公式也可以倒过来从没有使用Hadamard乘积而来的Wallis公式中得到$\zeta '\left( 0 \right)$的值【22】。

\begin{align*}\int_0^{\frac{\pi }{2}} {{{\sin }^n}xdx}  = \int_0^{\frac{\pi }{2}} {{{\cos }^n}xdx}  = \frac{{\sqrt \pi  \Gamma \left( {\frac{{n + 1}}{2}} \right)}}{{n\Gamma \left( {\frac{n}{2}} \right)}} = \left\{ \begin{array}{l}\frac{\pi }{2} \cdot \frac{{\left( {n - 1} \right)!!}}{{n!!}}&&n\text{为偶数时}\\\frac{{\left( {n - 1} \right)!!}}{{n!!}}&&n\text{为奇数时},\end{array} \right.\end{align*}

2.引理

${x^{b - a}}\frac{{\Gamma \left( {x + a} \right)}}{{\Gamma \left( {x + b} \right)}} = 1 + \frac{{\left( {a - b} \right)\left( {a + b - 1} \right)}}{{2x}} + O\left( {\frac{1}{{{x^2}}}} \right),x \to \infty .\tag{11}$

3.主要结果

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