Wallis' sequence estimated accurately using an alternating series

详细信息查看全文

作者：Vito Lampret ^{vito.lampret@guest.arnes.si" class="auth_mail" title="E-mail the corresponding author}
关键词：primary ; 26D20 ; 40A25 ; 41A60 ; secondary ; 11Y60 ; 11Y99 ; 65B15
刊名：Journal of Number Theory
出版年：2017
出版时间：March 2017
年：2017
卷：172
期：Complete
页码：256-269
全文大小：394 K

文摘

An asymptotic approximation of Wallis' sequence

m \mapsto W_{m} : = \prod_{k = 1}^{m} \frac{4 k^{2}}{4 k^{2} - 1}

is presented as

W_{m} = \frac{m π}{2 m + 1} \exp minsize="2.4ex">(2 σ_{q} (m) minsize="2.4ex">) \cdot \exp minsize="2.4ex">(r_{q} (m) minsize="2.4ex">),

where

\begin{matrix} σ_{q} (x) : = \sum_{i = 1}^{⌊ q / 2 ⌋} \frac{(1 - 4^{- i}) B_{2 i}}{i (2 i - 1) \cdot x^{2 i - 1}} \\ (B_{k} are the Bernoulli coefficients), \end{matrix}

and where

\begin{matrix} | r_{q} (m) | < r_{q}^{⁎} (m) : = \frac{2 π (q - 2)!}{3 {(2 m π)}^{q - 1}}, \\ for any integers m \geq 1 and q \geq 2 . \end{matrix}

Parameters m and q control the error factor exp⁡(r_q(m))

\exp minsize="2.4ex">(r_{q} (m) minsize="2.4ex">)

地址：北京市海淀区学院路29号邮编：100083

电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700