Shapiro Polynomials - Construction

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 a₀ = 1, a₁ = 1, a₂ = 1, a₃ = −1, etc.

The first Shapiro P_n(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 {a_n} has a fractal-like structure – for example, a_n = a_2n – which implies that the subsequence (a₀, a₂, a₄, ...) 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-)nz2n-1P_n(-1/z), or by the recursions