MATHEMATICS OF COMPUTATION Volume 74, Number 252, Pages 1923–1935 S 0025-5718(05)01736-9 Article electronically published on March 15, 2005 EVEN MOMENTS OF GENERALIZED RUDIN–SHAPIRO POLYNOMIALS CHRISTOPHE DOCHE Abstract. We know from Littlewood (1968) that the moments of order 4 of the classical Rudin–Shapiro polynomials P n (z) satisfy a linear recurrence of degree 2. In a previous article, we developed a new approach, which enables us to compute exactly all the moments M q (P n ) of even order q for q  32. We were also able to check a conjecture on the asymptotic behavior of M q (P n ), namely M q (P n ) ∼ C q 2 nq/2 , where C q =2 q/2 /(q/2 + 1), for q even and q  52. Now for every integer ℓ  2 there exists a sequence of generalized Rudin–Shapiro polynomials, denoted by P (ℓ) 0,n (z). In this paper, we extend our earlier method to these polynomials. In particular, the moments M q (P (ℓ) 0,n ) have been completely determined for ℓ = 3 and q =4, 6, 8, 10, for ℓ =4 and q =4, 6 and for ℓ =5, 6, 7, 8 and q = 4. For higher values of ℓ and q, we formulate a natural conjecture, which implies that M q (P (ℓ) 0,n ) ∼ C ℓ,q ℓ nq/2 , where C ℓ,q is an explicit constant. 1. Introduction Let T = {z ∈ C ||z| =1} be the complex torus and let f be an L q (T)–function. The moment of order q ∈ N of f satisfies (1) M q (f )=  1 0   f ( e 2iπt )  q dt. The Rudin–Shapiro polynomials [8] are defined by the recurrence relations (2) P n+1 (z)= P n (z)+ z 2 n Q n (z), Q n+1 (z)= P n (z) − z 2 n Q n (z) and the first values P 0 (z)= Q 0 (z) = 1. Obviously M 2 (P n )=2 n . In 1968, Littlewood [5] evaluated M 4 (P n ) and established that M 4 (P n ) ∼ 4 n+1 /3. In 1980, Saffari [7] conjectured that M q (P n ) ∼ 2 (n+1)q/2 q/2+1 · In [2] we were able to prove this result for q even less than or equal to 52. Received by the editor March 6, 2004 and, in revised form, May 31, 2004. 2000 Mathematics Subject Classification. Primary 11B83, 11B37, 42C05. Key words and phrases. Rudin–Shapiro polynomials, signal theory, Krawtchouk polynomials. c 2005 American Mathematical Society 1923 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use