The norm ratio of the polynomials with coefficients as binary sequence (Q1429346)

The author studies a problem connected with Rudin-Shapiro polynomials, inductively defined by \(P_0=Q_0=1\) and \[ P_n(e^{it})=P_{n-1}(e^{it})+e_{2^{n-1}} Q_{n-1}(e^{it}),\;n\geq 1, \] \[ P_n(e^{it})=P_{n-1}(e^{it})-e_{2^{n-1}} Q_{n-1}(e^{it}),\;n\geq 1, \] with \(e_k(t)=e^{ikt}\). The \(P_n\) have the form of a `binary polynomial' \[ P_n(e^{it})=\sum_{k=0}^{2^n-1}\, \epsilon_k e_k(t),\;\epsilon_k=\pm 1. \] The main result concerns the partial proof of the conjecture \[ \lim_{n\rightarrow\infty}\, {| | P_n| | _{2q}^{2q} \over | | P_n| | _q^{2q}} = {2^q\over q+1},\;\;q\text{ a positive integer}.\tag{\(*\)} \] It is shown that \((\ast)\) is correct for \(q=2\) and, moreover, that if \((\ast)\) holds for an \textit{even} \(q\), then it is true for \(q+1\).