Let $s(m)$ denote the number of distinct powers of 2 in the binary representation of $m$. Thus the Thue-Morse sequence is $(-1)^{s(m)}$ and
$T_n(x)=\sum_{0\leq m< 2^n}(-1)^{s(m)}e(mx)=\prod_{0\leq r< n}(1-e(2^rx))$
is a trigonometric generating generating function of the sequence. The work of Mauduit and Rivat on $(-1)^{s(p)}$ depends on nontrivial bounds for $\left \| T_n \right \|_1$ and for $\left \| T_n \right \|_\infty $. We consider other norms of the $T_n$. For positive integers $k$ let
$M_k(n)=\int_{0}^{1}\left | T_n(x) \right |^{2k}dx$
We show that the sequence $M_k(n)$ satisfies a linear recurrence of order $k$. Moreover, we determine a $k\times k$ matrix whose characteristic polynomial determines this linear recurrence.
This is joint work with Mauduit and Rivat.[-]