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.^[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:

{\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.

^ John Brillhart and L. Carlitz (May 1970). "Note on the Shapiro polynomials". Proceedings of the American Mathematical Society. 25 (1). Proceedings of the American Mathematical Society, Vol. 25, No. 1: 114–118. doi:10.2307/2036537. JSTOR 2036537.
^ Somaini, U. (June 26, 1975). "Binary sequences with good correlation properties". Electronics Letters. 11 (13): 278–279. Bibcode:1975ElL....11..278S. doi:10.1049/el:19750211. Archived from the original on February 26, 2019.