Even moments of generalized Rudin-shapiro polynomials

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_q2^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_qP_0,n^(ℓ)) ∼ C _ℓ,qℓ^nq/2, where C_ℓ,q is an explicit constant.