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Shapiro polynomials

In mathematics, the Shapiro polynomials are a sequence of polynomials which were first studied by Harold S. Shapiro in 1951 when considering the magnitude of specific trigonometric sums.[1] In signal processing, the Shapiro polynomials have good autocorrelation properties[2] and their values on the unit circle are small. The first few members of the sequence are:

\begin{align}P_1(x) & {} =1 + x \\P_2(x) & {} =1 + x + x^2 - x^3 \\P_3(x) & {} =1 + x + x^2 - x^3 + x^4 + x^5 - x^6 + x^7 \\... \\Q_1(x) & {} =1 - x \\Q_2(x) & {} =1 + x - x^2 + x^3 \\Q_3(x) & {} =1 + x + x^2 - x^3 - x^4 - x^5 + x^6 - x^7 \\... \\\end{align}

where the second sequence, indicated by Q, is said to be complementary to the first sequence, indicated by P.

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