Résumé : 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
$T_n(x)=\sum _{0\le m<2^n}(-1{)}^{s(m)}e(mx)=\prod _{0\le r<n}(1-e(2^{r}x))$
is a trigonometric generating generating function of the sequence. The work of Mauduit and Rivat on $(-1{)}^{s(p)}$ depends on nontrivial bounds for ${\Vert T_n\Vert }_1$ and for ${\Vert T_n\Vert }_{\mathrm{\infty }}$. We consider other norms of the $T_n$. For positive integers $k$ let
$M_k(n)={\int }_0^1{|T_n(x)|}^{2k}dx$
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.