By Shalosh B. Ekhad and Doron Zeilberger
[Appeared in "Analytic number theory, modular forms and q-hypergeometric series"( in honor of Krishna Alladi, F. Garvan, ed.), 253-265, Springer Proc. Math. Stat., 221, Springer, Cham, 2017.]
Previous version: May 27, 2016
Last update (of webpage, and article): June 7, 2016
Dedicated to Krishnaswami "Krishna" Alladi, the tireless apostle of Srinivasa Ramanujan, yet a great mathematician on his own right.
In particular, I really enjoyed Hugh Montgomery's Erdős Colloquium talk about the Rudin-Shapiro polynomials, that lead to the present article, Maple package, and ample output.
Abstract: Continuing pioneering work of Christophe Doche and Laurent Habsieger from 2004, we develop computer algebra algorithms, implemented in Maple, for finding the (necessarily rational) generating function for any integral of products, and in particular, moments, of Rudin-Shapiro polynomials. We generate a lot of output, and confirm again a conjecture of Saffari for the asymptotics for small (and not so small) powers. We also confirm, for small powers, a related, more general, conjecture, of Hugh Montgomery. Finally, we outline a proof of Saffari's full conjecture, that we believe can be turned into a full proof.
Added May 23, 2016: Brad Rodgers, independently, and simultaneously, found a (complete) proof of Saffari's conjecture, that he is writing up now and will soon post in the arxiv. Meanwhile, you can read his proof in a letter that he wrote one of us (DZ).
Added May 30, 2016: Tamas Erdelyi has an interesting article where he examines the Mahler measure of the Rudin-Shapiro polynomials. This turns out to be closely related to the moments of the Rudin-Shapiro polynomials, (see Theorem 2.2), from which an upper bound
M_{n}(k) < c((2^n)/(\log n))*(2^{kn})
follows immediately. This is far from Saffari's Conjecture but it is quite non-trivial.
Added June 7, 2016: Brad Rodgers' beautiful article has just been posted. He also proves the more general Montgomery conjecture. Highly recommended.
[You also need to download, in the same directory, the data file ShapiroData.txt.]
P_{k}(z)
that are defined by a functional recurrence of the form
P_{k}(z)= C_{1}(z) P_{k-1}(z^{r})+ C_{2}(z) P_{k-1}(-z^{r})+ C_{3}(z) P_{k-1}(z^{-r})+ C_{4}(z) P_{k-1}(-z^{-r})
where r is a positive integer ≥ 2, and C_{1}(z),C_{2}(z),C_{3}(z),C_{4}(z)
are Laurent polynomials in z
(whose degree is < r and low-degree > -r). The original case correspods to
r=2, C_{1}(z)=1, C_{2}(z)=z, C_{3}(z)=0, C_{4}(z)=0,
If you want to see explicit generation functions for the sequences of even moments from the second (easy!) through the 20th (not so easy!) of the famous Rudin-Shapiro polynomials, as well as the first 101 terms for each sequence, as well as rigorous confirmation (for these cases, i.e. up to the 20th moment) of Bahman Saffari's conjecture
If you want to see explicit generation functions for the sequences of integrals defined by
∫_{0}^{1} P_{k}(e(θ))^{m}P_{k}(e(-θ))^{n} dθ
(where, as usual e(θ)=e^{2πiθ}, and P_{k}(z) are the Rudin-Shapiro polynomials), for 1 ≤ m < n ≤ 7, the first 41 terms of each of these sequnces, and a rigorous proof (for the above values of m and n) of Hugh Montgomery's conjecture made in Oberwolfach see pp. 67-68 here,
∫_{0}^{1} P_{k}(e(θ))^{m}P_{k}(e(-θ))^{n} dθ
(where, as usual e(θ)=e^{2πiθ}, and P_{k}(z) are the Rudin-Shapiro polynomials), for 1 ≤ m < n ≤ 10, and a numerical confirmation (for the above values of m and n) of Hugh Montgomery's conjecture made in Oberwolfach see pp. 67-68 here,
If you want to see rigorous confirmation of the fact that the heaviest words in {a,b,A,B} are all of the form (aA)^{m-i}(bB)^{i}, for i from 0 to m, thereby justifying our tentative proof of Saffari's conjecture
If you want to see the first 101 terms for each of the sequences defined by
∫_{0}^{1} (P_{k}(e(θ))P_{k}(e(-θ)))^{m-i} (P_{k}(-e(θ))P_{k}(-e(-θ)))^{i} dθ
for 1 ≤ m ≤ 8, and 0 ≤ i ≤ m/2, and a numerical confirmation, for these cases, of our generalization of Saffari's conjecture
P_{k}(z)= (-9/z+1+5z) P_{k-1}(z^{2})+ (5+7z) P_{k-1}(-z^{2})+ (17/z+19z) P_{k-1}(z^{-2})+ (1/z+3z) P_{k-1}(-z^{-2}) , P_{0}(z)=1
P_{k}(z)= P_{k-1}(z^{3})+ 3z P_{k-1}(-z^{3})+ (z+1/z^2) P_{k-1}(z^{-3})+ (1/z+1+3z^{2}) P_{k-1}(-z^{-3}) , P_{0}(z)=1