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

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.

我们探索了无限乘积的评估，涉及自动序列\((d_{n}: n \in \mathbb {N}_{0})\) ，称为倍周期序列，受到 Allouche、Riasat 工作的启发，以及 Shallit 对涉及 Thue-Morse 或 Golay-Shapiro 序列的无限乘积的评估。我们的方法允许应用积分运算符，从而导致涉及双对数函数的表达式的新乘积展开，从而产生涉及加泰罗尼亚常数G的新公式，例如公式

本文介绍了。更一般地，对于初等函数e ( n ) 形式为\( \prod _{n=1}^{\infty } e(n)^{d_{n}} \) 的无限乘积的求值是主要的我们文章的目的。过去涉及自动序列的无限乘积的工作主要涉及自动序列a ( n ) 和有理函数R ( n )，与上面显示的产品评估中的结果形成对比。我们的方法还允许我们获得涉及涉及倍周期序列的无限乘积的\(\frac{\zeta (3)}{\pi ^2}\) 的新评估。