Transcendental Infinite Products Associated with the $\pm 1$ Thue-Morse Sequence

TL;DR

This paper establishes new infinite product identities involving the Thue-Morse sequence and fundamental constants like pi and sqrt(2), expanding understanding of their algebraic properties and connections to the Flajolet-Martin constant.

Contribution

It introduces three novel infinite product identities linked to the Thue-Morse sequence and derives a new expression related to the Flajolet-Martin constant.

Findings

New identities involving pi and sqrt(2)

Enhanced understanding of Thue-Morse sequence properties

A novel expression for a product related to the Flajolet-Martin constant

Abstract

Infinite products associated with the $\pm 1$ Thue-Morse sequence whose value is rational or algebraic irrational have been studied by several authors. In this short note we prove three new infinite product identities involving ${\pi}$, $\sqrt{2}$, and the $\pm 1$ Thue-Morse sequence, building on a result by Allouche, Riasat, and Shallit. We then use our method to find a new expression for a product appearing within the context of the Flajolet-Martin constant.