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 $\mathcal{M}_q({P_n})$ of even order $q$ for $q\leqslant 32$. We were also able to check a conjecture on the asymptotic behavior of $\mathcal{M}_q({P_n})$, namely $\mathcal{M}_q({P_n})\sim C_q 2^{nq/2}$, where $C_q = 2^{q/2}/(q/2+1)$, for $q$ even and $q\leqslant 52$. Now for every integer $\ell\geqslant 2$ there exists a sequence of generalized Rudin-Shapiro polynomials, denoted by $P_{0,n}^{(\ell)}(z)$. In this paper, we extend our earlier method to these polynomials. In particular, the moments $\mathcal{M}_q(P_{0,n}^{(\ell)})$ have been completely determined for $\ell=3$ and $q=4,6,8,10$, for $\ell=4$ and $q=4,6$ and for $\ell = 5,6,7,8$ and $q=4$. For higher values of $\ell$ and $q$, we formulate a natural conjecture, which implies that $\mathcal{M}_q(P_{0,n}^{(\ell)})\sim C_{\ell,q}\ell^{nq/2}$, where $C_{\ell,q}$ is an explicit constant.