## Abstract

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 M_{q}(P_{n}) of even order q for q ≤ 32. We were also able to check a conjecture on the asymptotic behavior of M_{q}(P_{n}), namely M_{q}(P_{n}) ∼ C_{q}2^{nq/2}, where C_{q}, = 2^{q/2}/(q/2 + 1), for q even and q ≤ 52. Now for every integer ℓ ≥ 2 there exists a sequence of generalized Rudin-Shapiro polynomials, denoted by P _{0,n}^{(ℓ)}, (z). In this paper, we extend our earlier method to these polynomials. In particular, the moments M_{q},(P _{0,n}^{(ℓ)} have been completely determined for ℓ= 3 and q = 4, 6, 8, 10, for ℓ = 4 and 9 = 4, 6 and for ℓ= 5, 6, 7, 8 and q = 4. For higher values of ℓ and q, we formulate a natural conjecture, which implies that_{q}P_{0,n}^{(ℓ)}) ∼ C _{ℓ,q}ℓ^{nq/2}, where C_{ℓ,q} is an explicit constant.

Original language | English |
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Pages (from-to) | 1923-1935 |

Number of pages | 13 |

Journal | Mathematics of Computation |

Volume | 74 |

Issue number | 252 |

DOIs | |

Publication status | Published - Oct 2005 |