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.