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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 and their values on the unit circle are small.[2] The first few members of the sequence are:

The Shapiro polynomials Pn(z) may be constructed from the Golay–Rudin–Shapiro sequence an, which equals 1 if the number of pairs of consecutive ones in the binary expansion of n is even, and −1 otherwise. Thus a0 = 1, a1 = 1, a2 = 1, a3 = −1, etc.

The first Shapiro Pn(z) is the partial sum of order 2n − 1 (where n = 0, 1, 2, ...) of the power series

The Golay–Rudin–Shapiro sequence {an} has a fractal-like structure – for example, an = a2n – which implies that the subsequence (a0, a2, a4, ...) replicates the original sequence {an}. This in turn leads to remarkable functional equations satisfied by f(z).

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