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Single angle Moments of a Thue-Morse generating function

Auteurs : Montgomery, Hugh L. (Auteur de la Conférence)
CIRM (Editeur )

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    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
    $T_n(x)=\sum_{0\leq m< 2^n}(-1)^{s(m)}e(mx)=\prod_{0\leq r< n}(1-e(2^rx))$
    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.

    11B83 - Special sequences and polynomials

    Informations sur la rencontre

    Nom du congrès : Prime numbers : new perspectives / Nombres premiers : nouvelles perspectives
    Organisteurs Congrès : Dartyge, Cécile ; Mauduit, Christian ; Rivat, Joël ; Stoll, Thomas
    Dates : 10/02/14 - 14/02/14
    Année de la rencontre : 2014

    Citation Data

    DOI : 10.24350/CIRM.V.18610603
    Cite this video as: Montgomery, Hugh L. (2014). Moments of a Thue-Morse generating function.CIRM . Audiovisual resource. doi:10.24350/CIRM.V.18610603
    URI : http://dx.doi.org/10.24350/CIRM.V.18610603