Let $T$ be the generating function of the [Thue-Morse sequence](https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_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!



Generating function of the Thue-Morse sequence
Let 
$$F(x)=1-x-x^2+x^3-x^4+x^5+\ldots=(1-x)(1-x^2)(1-x^4)\ldots.$$
Then $F(x)=(1-x)F(x^2)$ and
$$F(x)=1+x+x^2+x^3+\ldots-2T(x)=\frac{1}{1-x}-2T(x).$$
So
$$T(x)-(1-x)T(x^2)=\frac{x}{1-x^2}.$$


