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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Even moments of generalized Rudin–Shapiro polynomials
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by Christophe Doche PDF
Math. Comp. 74 (2005), 1923-1935 Request permission


We know from Littlewood (1968) that the moments of order $4$ of the classical Rudin–Shapiro polynomials $P_n(z)$ satisfy a linear recurrence of degree $2$. In a previous article, we developed a new approach, which enables us to compute exactly all the moments $\mathcal {M}_q({P_n})$ of even order $q$ for $q\leqslant 32$. We were also able to check a conjecture on the asymptotic behavior of $\mathcal {M}_q({P_n})$, namely $\mathcal {M}_q({P_n})\sim C_q 2^{nq/2}$, where $C_q = 2^{q/2}/(q/2+1)$, for $q$ even and $q\leqslant 52$. Now for every integer $\ell \geqslant 2$ there exists a sequence of generalized Rudin–Shapiro polynomials, denoted by $P_{0,n}^{(\ell )}(z)$. In this paper, we extend our earlier method to these polynomials. In particular, the moments $\mathcal {M}_q(P_{0,n}^{(\ell )})$ have been completely determined for $\ell =3$ and $q=4$, 6, 8, 10, for $\ell =4$ and $q=4$, 6 and for $\ell = 5$, 6, 7, 8 and $q=4$. For higher values of $\ell$ and $q$, we formulate a natural conjecture, which implies that $\mathcal {M}_q(P_{0,n}^{(\ell )})\sim C_{\ell ,q}\ell ^{nq/2}$, where $C_{\ell ,q}$ is an explicit constant.
  • C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, User’s guide to PARI / GP, ver. 2.1.3, 2002.
  • C. Doche and L. Habsieger, Moments of the Rudin–Shapiro polynomials. Journal of Fourier Analysis and Applications 10 (2004).
  • Ilia Krasikov and Simon Litsyn, On integral zeros of Krawtchouk polynomials, J. Combin. Theory Ser. A 74 (1996), no. 1, 71–99. MR 1383506, DOI 10.1006/jcta.1996.0038
  • Zhi Xian Lei, Some properties of generalized Rudin-Shapiro polynomials, Chinese Ann. Math. Ser. A 12 (1991), no. 2, 145–153 (Chinese). MR 1112426
  • John E. Littlewood, Some problems in real and complex analysis, D. C. Heath and Company Raytheon Education Company, Lexington, Mass., 1968. MR 0244463
  • A GP–PARI program to compute the moments of some generalized Rudin–Shapiro polynomials.
  • B. Saffari, personal communication, (2001).
  • H. S. Shapiro, Extremal problems for polynomials and power series, Ph.D. thesis, MIT, 1951.
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Additional Information
  • Christophe Doche
  • Affiliation: Division of ICS, Building E6A, Macquarie University, New South Wales 2109 Australia
  • Email:
  • Received by editor(s): March 6, 2004
  • Received by editor(s) in revised form: May 31, 2004
  • Published electronically: March 15, 2005
  • © Copyright 2005 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 1923-1935
  • MSC (2000): Primary 11B83, 11B37, 42C05
  • DOI:
  • MathSciNet review: 2164103