  Letting $(t_n)$ denote the Thue-Morse sequence with values $0, 1$, we note
that the Woods-Robbins product $$ \prod_{n \geq 0}
\left(\frac{2n+1}{2n+2}\right)^{(-1)^{t_n}} = 2^{-1/2} $$ involves a rational
function in $n$ and the $\pm 1$ Thue-Morse sequence $((-1)^{t_n})_{n \geq 0}$.
The purpose of this paper is twofold. On the one hand, we try to find other
rational functions for which similar infinite products involving the $\pm 1$
Thue-Morse sequence have an expression in terms of known constants. On the
other hand, we also try to find (possibly different) rational functions $R$ for
which the infinite product $\prod R(n)^{t_n}$ also has an expression in terms
of known constants.
