  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.
