Infinite products involving binary digit sums

TL;DR

This paper investigates infinite products involving the Thue-Morse sequence, deriving new identities and analyzing their properties to extend understanding of such products beyond the classical Woods-Robbins identity.

Contribution

The paper introduces new identities for infinite products with the Thue-Morse sequence and analyzes their analytical properties, expanding the known results in this area.

Findings

Derived new identities similar to Woods-Robbins product

Analyzed the function f(b,c) involving binary digit sums

Extended understanding of infinite products with Thue-Morse sequence

Abstract

Let $(u_n)_{n\ge 0}$ denote the Thue-Morse sequence with values $\pm 1$. The Woods-Robbins identity below and several of its generalisations are well-known in the literature \begin{equation*}\label{WR}\prod_{n=0}^\infty\left(\frac{2n+1}{2n+2}\right)^{u_n}=\frac{1}{\sqrt 2}.\end{equation*} No other such product involving a rational function in $n$ and the sequence $u_n$ seems to be known in closed form. To understand these products in detail we study the function \begin{equation*}f(b,c)=\prod_{n=1}^\infty\left(\frac{n+b}{n+c}\right)^{u_n}.\end{equation*} We prove some analytical properties of $f$. We also obtain some new identities similar to the Woods-Robbins product.