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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!

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1 Answer 1

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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 $$T(x)-(1-x)T(x^2)=\frac{x}{1-x^2}.$$

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    $\begingroup$ Thanks! BTW, an immediate corollary of this identity is the congruence I've mentioned in my post. $\endgroup$
    – Seva
    Oct 5, 2015 at 8:16
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    $\begingroup$ @Seva If you'll find another equation on $T$ then we'll have another identity for $F$. $\endgroup$ Oct 5, 2015 at 11:43

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