Even moments of generalized Rudin--Shapiro polynomials

Even moments of generalized Rudin-Shapiro polynomials

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) ∼ Cq2, where Cq = 2...

Moments of the Rudin-Shapiro Polynomials

We develop a new approach of the Rudin-Shapiro polynomials. This enables us to compute their moments of even order q for q 32, and to check a conjecture on the asymptotic behavior of these moments for q even and q 52.

Generalized Rudin – Shapiro sequences

Crosscorrelation of Rudin-Shapiro-Like Polynomials

We consider the class of Rudin-Shapiro-like polynomials, whose L norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial f(z) = f0 + f1z + · · ·+ fdz d is identified with the sequence (f0, f1, . . . , fd) of its coefficients. From the L 4 norm of a polynomial, one can easily calculate the autocorrelation merit factor of its associated sequence, and conversely. I...

Rudin-Shapiro-like polynomials in L4

We examine sequences of polynomials with {+1,−1} coefficients constructed using the iterations p(x) → p(x) ± xd+1p∗(−x), where d is the degree of p and p∗ is the reciprocal polynomial of p. If p0 = 1 these generate the Rudin-Shapiro polynomials. We show that the L4 norm of these polynomials is explicitly computable. We are particularly interested in the case where the iteration produces sequenc...