Even moments of generalized Rudin-shapiro polynomials

Christophe Doche

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5 Citations (Scopus)


We know from Littlewood (1968) that the moments of order 4 of the classical Rudin-Shapiro polynomials Pn(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 Mq(Pn) of even order q for q ≤ 32. We were also able to check a conjecture on the asymptotic behavior of Mq(Pn), namely Mq(Pn) ∼ Cq2nq/2, where Cq, = 2q/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 Mq,(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 thatqP0,n(ℓ)) ∼ C ℓ,qnq/2, where Cℓ,q is an explicit constant.

Original languageEnglish
Pages (from-to)1923-1935
Number of pages13
JournalMathematics of Computation
Issue number252
Publication statusPublished - Oct 2005


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