Powers of the Thue--Morse Series: 2-Adic Valuations and Automatic Odd Parts

Let $T(x)={\prod }_{j\ge 0}(1-x^{2^j})={\sum }_{n\ge 0}t(n)x^n$ be the Thue–Morse generating function, and write $T(x{)}^m={\sum }_{n\ge 0}{t}_m(n)x^n$ for a positive integer $m$. For a nonzero integer $a$, let ${\nu }_2(a)$ be the exponent of $2$ in $a$ and put $\mathrm{odd}(a)=a\mathrm{/}{2}^{{\nu }_2(a)}$. For every $s\ge 1$, we prove that the sequence $(\mathrm{odd}(t_m(n))2^s{)}_{n\ge 0}$ is $2$-automatic when $m=2^r$, $r\ge 1$, or $m=3\cdot 2^r$, $r\ge 2$. In both families, ${\nu }_2(t_m(n))={\nu }_2\left(\genfrac{}{}{0px}{}{n+m-1}{m-1}\right)$. The valuation identity for $m=2^r$ is known; our proof recovers it and also establishes the automaticity assertion. For $m=6$ we prove ${\nu }_2(t_6(n))={\nu }_2\left(\genfrac{}{}{0px}{}{n+5}{5}\right)+{\mathbf{1}}_{n\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)}$ and show that its odd parts modulo every $2^s$ are $2$-automatic.