包含倍周期序列的无穷积

John M. Campbell
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引用次数: 0

摘要

我们从Allouche, Riasat和Shallit对涉及Thue-Morse或Golay-Shapiro序列的无限积的评估工作的启发,探索了涉及自动序列\((d_{n}: n \in \mathbb {N}_{0})\)的无限积的评估,称为倍周期序列。我们的方法允许应用积分运算符,对涉及二重函数的表达式进行新的乘积展开,从而产生涉及加泰罗尼亚常数G的新公式,例如本文中介绍的公式$$\begin{aligned} \prod _{n=1}^{\infty } \left( \left( \frac{n+2}{n}\right) ^{n+1} \left( \frac{4 n + 3}{4 n+5}\right) ^{4 n+4}\right) ^{d_{n}} = \frac{e^{\frac{2 G}{\pi }}}{\sqrt{2}} \end{aligned}$$。更一般地说,对于初等函数e(n),求形式为\( \prod _{n=1}^{\infty } e(n)^{d_{n}} \)的无穷积的值是本文的主要目的。过去关于涉及自动序列的无穷积的工作主要是关于自动序列a(n)和有理函数R(n)的乘积\( \prod _{n=1}^{\infty } R(n)^{a(n)} \)的形式,与我们的结果相反,如上所示的产品评估。我们的方法还允许我们获得涉及\(\frac{\zeta (3)}{\pi ^2}\)的涉及周期加倍序列的无限乘积的新评估。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Infinite products involving the period-doubling sequence

We explore the evaluation of infinite products involving the automatic sequence \((d_{n}: n \in \mathbb {N}_{0})\) known as the period-doubling sequence, inspired by the work of Allouche, Riasat, and Shallit on the evaluation of infinite products involving the Thue–Morse or Golay–Shapiro sequences. Our methods allow for the application of integral operators that result in new product expansions for expressions involving the dilogarithm function, resulting in new formulas involving Catalan’s constant G, such as the formula

$$\begin{aligned} \prod _{n=1}^{\infty } \left( \left( \frac{n+2}{n}\right) ^{n+1} \left( \frac{4 n + 3}{4 n+5}\right) ^{4 n+4}\right) ^{d_{n}} = \frac{e^{\frac{2 G}{\pi }}}{\sqrt{2}} \end{aligned}$$

introduced in this article. More generally, the evaluation of infinite products of the form \( \prod _{n=1}^{\infty } e(n)^{d_{n}} \) for an elementary function e(n) is the main purpose of our article. Past work on infinite products involving automatic sequences has mainly concerned products of the form \( \prod _{n=1}^{\infty } R(n)^{a(n)} \) for an automatic sequence a(n) and a rational function R(n), in contrast to our results as in above displayed product evaluation. Our methods also allow us to obtain new evaluations involving \(\frac{\zeta (3)}{\pi ^2}\) for infinite products involving the period-doubling sequence.

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