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 q 2 nq/2 , where C q = 2 q/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 M q (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 M q (P () 0,n ) ∼ C ,q nq/2 , where C ,q is an explicit constant.