Infinite products involving the period-doubling sequence

Abstract

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.