Infinite products involving binary digit sums

Samin Riasat

Published 2017-09-13Version 1

Let $(u_n)_{n\ge 0}$ be the $\pm1$ Thue-Morse sequence; i.e., $u_n$ is equal to $1$ if the binary expansion of $n$ has an even number of $1$'s, and is equal to $-1$ otherwise. The following identity, originally due to Woods and Robbins, 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*} But 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 more identities similar to the Woods-Robbins product, such as \begin{equation*}\prod_{n=0}^\infty\left(\frac{4n+1}{4n+3}\right)^{u_n}=\frac 12\end{equation*} \begin{equation*}\label{}\prod_{n=0}^\infty\left(\frac{(2n+2)(n+1)}{(2n+3)(n+2)}\right)^{u_n}=\frac{1}{\sqrt 2}.\end{equation*}