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 = 2q/2/(q/2 + 1), for q even and q 52. Now for every integer 2 there exists a sequence of generalized Rudin–Shapiro polynomials, denoted by P ( ) 0,n(z). In this paper, we extend our earlier method to these polynomials. In particular, the moments Mq(P ( ) 0,n) have been completely determined for = 3 and q = 4, 6, 8, 10, for = 4 and q = 4, 6 and for = 5, 6, 7, 8 and q = 4. For higher values of and q, we formulate a natural conjecture, which implies that Mq(P ( ) 0,n) ∼ C ,q nq/2, where C ,q is an explicit constant.