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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