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. In signal processing, the Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. The first few members of the sequence are:
where the second sequence, indicated by "Q", is said to be "complementary" to the first sequence, indicated by "P".
Construction.
The Shapiro polynomials "P""n"("z") may be constructed from the Golay–Rudin–Shapiro sequence "a""n", which equals 1 if the number of pairs of consecutive ones in the binary expansion of "n" is even, and −1 otherwise. Thus "a"0 = 1, "a"1 = 1, "a"2 = 1, "a"3 = −1, etc.
The first Shapiro "P""n"("z") is the partial sum of order 2"n" − 1 (where "n" = 0, 1, 2, ...) of the power series
The Golay–Rudin–Shapiro sequence {"a""n"} has a fractal-like structure – for example, "a""n" = "a"2"n" – which implies that the subsequence ("a"0, "a"2, "a"4, ...) replicates the original sequence {"a""n"}. This in turn leads to remarkable
functional equations satisfied by "f"("z").
The second or complementary Shapiro polynomials "Q""n"("z") may be defined in terms of this sequence, or by the relation "Q""n"("z") = (1-)"n""z"2"n"-1"P""n"(-1/"z"), or by the recursions
Properties.
The sequence of complementary polynomials "Q""n" corresponding to the "P""n" is uniquely characterized by the following properties:
The most interesting property of the {"P""n"} is that the absolute value of "P""n"("z") is bounded on the unit circle by the square root of 2("n" + 1), which is on the order
of the L2 norm of "P""n". Polynomials with coefficients from the set {−1, 1} whose maximum modulus on the unit circle is close to their mean modulus are useful for various applications in communication theory (e.g., antenna design and data compression). Property (iii) shows that ("P", "Q") form a Golay pair.
These polynomials have further properties: