  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.
