Report on 
"Integrals Involving Rudin-Shapiro Polynomials and Sketch
of a Proof of Saffari's Conjecture" 
by Shalosh B. Ekhad and Doron Zeilberger
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This paper is about the computation of moments of the Rudin-Shapiro
polynomials P_k(z), which are defined by the linear functional
recurrence equation

  P_k(z) = P_{k-1}(z^2) + z*P_{k-1}(-z^2),  P_0(z) = 1,

and which are of interest in signal processing due to their special
autocorrelation properties.  These moments, denoted by M_n(k),
can be either defined by a certain integral over the unit interval, 
or as the constant term of the Laurent polynomial (P_k(z)*P_k(1/z))^n; 
in the presented approach the second definition is employed, 
which leads to the remarkable fact that there is, other than 
suggested by the title, not a single integral sign appearing in
the article.  In the first part of the paper, an algorithmic 
approach to compute the generating function of the n-th moments,

  R_n(t) = \sum_{k>=0} M_n(k)*t^k,

is developed. This approach, to some extent, follows the work of
[DH], and reproduces/confirms their results. While [DH]'s approach is 
tailored to the specific setting of Rudin-Shapiro polynomials, 
the article under review presents a more algorithmic method 
that can be easily generalized to other families of polynomials. 
Interestingly, each R_n(t) is a rational function, and the first members 
(n <= 5) are given in the paper, and some more (n <= 10) in the 
accompanying electronic material.

In the second part of the paper, the authors turn to Saffari's
conjecture, which concerns the asymptotic growth of M_n(k), 
the sequence of n-th moments.  A strategy for proving this conjecture is 
then discussed in detail, but it stays at the level of a "proof sketch" 
(as it is honestly announced in the title): Propositions 1 and 2 have been 
verified empirically for many instances but have not been proven rigorously. 
Also for the "small change hypothesis", on which the whole argument is based, 
only some evidence and proof sketch is given.  The paper concludes by 
mentioning that the presented methods can be applied to
other families of polynomials that are defined by a linear
functional recurrence equation of similar type as the Rudin-Shapiro recurrence.

It is a pleasure to read this article, as all steps are well motivated and
well explained. The presented material and methods are very interesting.
Although Saffari's conjecture (and its generalization by Montgomery)
have been proven rigorously by Brad Rodgers (at the same time and
independently, see the addendum made on June 7), we believe that the 
present article is of independent interest. Hence I highly recommend 
to accept it for publication. The motivation for the present work 
arose from a lecture given at the conference to celebrate 
Alladi's 60th birthday; this makes the ALLADI60 conference proceedings 
a perfect place to publish this article. Some more
detailed comments follow.

Proposition 1: The method "holonomic ansatz" laid out in [Z] is
not applicable to this determinant, for two reasons: first, the matrix entries
K_n(m,r) depend on the parameter n (the dimension of the matrix), while the
holonomic ansatz requires that the (n-1)*(n-1)-matrix is a submatrix of the
n*n-matrix. Second, because the quotient det(zI-K_{n+1})/det(zI-K_n)
doesn't look like a holonomic sequence: note that the factors in the products
do not cancel since they depend on n themselves.

Proposition 2: it is puzzling to observe that the author is not
willing to apply the WZ proof theory (invented by Wilf and himself!) to this
binomial sum. What is the reason?

A remark concerning the finiteness of the rewriting scheme (from
which the rationality of the moment generating function follows): From the
linearity of (DefiningRecurrence) it follows that the total degree with 
respect to a, A, b, B always stays the same, and hence that there is 
only a finite set of monomials that can occur. The problem is the 
variable z which in principle could occur with arbitrarily high powers in the 
recursive process. So it is not completely obvious that this process 
terminates, but seemingly this has been proven by [DH] in the case of 
Rudin-Shapiro polynomials.  

On page 10 it is stated "One always gets a finite scheme", 
without any further explanation. Does the finiteness in the general case 
also follow from [DH]'s result? Or what is the justification/intuition 
for this claim?

In the statement of Saffari's conjecture, it would be helpful if
the symbol ~ ("asymptotically equivalent") came with an indication which 
variable tends to infinity (presumably k). Since the same notation 
appears several times later, it would be good to clarify 
(once, at first occurrence) that always k is meant.

page 3: squence -> sequence
page 4, line 3: use I for the identity matrix (as in Prop. 1)
page 4-5: the expressions for R_4 and R_5 would look nicer if ^ was used 
          instead of **
page 5: in the equation after "that we abbreviate" the CT is missing
page 8: one could add that the largest eigenvalue has multiplicity one.
page 8: equation after "Hence" should have ~ instead of =
page 9, 2nd paragraph, line 2: then -> them