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:
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 . . . {\displaystyle {\begin{aligned}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{aligned}}}
where the second sequence, indicated by Q, is said to be complementary to the first sequence, indicated by P.
Construction 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
f(z) := a0 + a1 z + a2 z2 + ... 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). The second or complementary Shapiro polynomials Qn(z) may be defined in terms of this sequence, or by the relation Qn(z) = (−1)nz2n−1Pn(−1/z), or by the recursions
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