Let $T$ be the generating function of the Thue-Morse sequence; thus, $T(x)=x+x^2+x^4+x^7+\dotsb$. It is known that $T$ satisfies the nice congruence $$ (1+x)^3 T^2(x) + (1+x)^2 T(x) + x \equiv 0 \pmod 2 $$ (the congruence is actually modulo the principal ideal generated by $2$ in the ring of formal power series ${\mathbb Z}[[x]]$). Does $T$ satisfy any identity of this or some other sort? Is there some product representation for $T$? In brief, I am interested in the properties of $T$ in the ring ${\mathbb Z}[[x]]$ itself rather than in its quotient rings.
Thanks in advance for any pointers!