Integrals Involving Rudin-Shapiro Polynomials and Sketch of a Proof of Saffari’s Conjecture

By Shalosh B. EKHAD and Doron ZEILBERGER

Dedicated to Krishnaswami “Krishna” Alladi, the tireless apostle of Srinivasa Ramanujan,

yet a great mathematician on his own right.

Preface: Krishna Alladi

One of the greatest disciples of Srinivasa Ramanujan, who did so much to make him a household name in the mathematical community, and far beyond, is Krishnaswami “Krishna” Alladi. Among many other things, he founded and is still editor-in-chief, of the very successful Ramanujan Journal (very ably managed by managing editor Frank Garvan), and initiated the SASTRA Ramanujan prize given to promising young mathematicians.

But Krishna is not just a mathematical leader, he is also a great number-theorist with very broad interests, including analytic number theory and, inspired by Ramanujan, q-series and partitions. That’s why it is not surprising that the conference to celebrate his 60th birthday, that took place last March, attracted attendees and speakers with very diverse interests, and enabled the participants to learn new things far afield from their own narrow specialty. That’s how we found out, and got hooked on, Rudin-Shapiro polynomials.

Hugh Montgomery’s Erdős’s colloquium

One of the highlights of the conference was a fascinating talk by the eminent Michigan number theorist (and Krishna’s former postdoc mentor) Hugh Montgomery, who talked about Littlewood polynomials of interest both in pure number theory and, surprisingly, in signal processing. These are polynomials whose coefficients are in {−1, 1}. Among these stand out the famous Rudin-Shapiro polynomials, introduced ([S1][S2]) by Harold “Silent” Shapiro^{1} and rediscovered by Walter Rudin ([R]).

The Rudin-Shapiro polynomials

The Rudin-Shapiro polynomials, P_{k}(z), are best defined by the functional recurrence (see [Wi])

P_{k}(z) = P_{k−1}(z^{2}) + zP_{k−1}(−z^{2}) ,  (DefiningRecurrence)

1 Harold S. Shapiro (S. originally stood for Seymour) was one of a brilliant cohort of students at City College, in the

late 1940s, that included Leon Ehrenpreis, Donald Newman, Israel Aumann, and another Harold Shapiro, Harold N. Shapiro (N. originally stood for Nathaniel). But their friends, in order to distinguish between the two Harold Shapiros, called them “Silent” and “Noisy” respectively. It is ironic that Harold Silent Shapiro’s son is the eminent, but very loud, MIT cosmologist, Max Tegmark.

with the initial condition P_{0}(z) = 1.

As Hugh Montgomery described so well in his talk, these have amazing properties. Both numbertheorists and signal-processors are very interested in the so-called sequence of (even) moments, whose definition usually involves the integral sign, but is better phrased entirely in terms of highschool algebra as follows.

M_{n}(k) := CT [P_{k}(z)^{n}P_{k}(z^{−1})^{n}] ,

where CT denotes the “constant term functional”, that for any Laurent polynomial f(z) of z, extracts the coefficient of z^{0}. For example CT [4/z^{2} + 11/z + 101 + 5z + 11z^{15}] = 101.

Can we find closed-form expressions for M_{n}(k), in k, for any given, specific, positive integer n? Failing this, can we find explicit expressions for the generating functions

R_{n}(t) := ∑_{k=0}^{∞} M_{n}(k)t^{k} ?

The sequence M_{1}(k) has a very nice closed-form, M_{1}(k) = 2^{k}. This is not very hard, even for humans. Indeed, using Eq. (DefiningRecurrence), we get

P_{k}(z)P_{k}(z^{−1}) = ( P_{k−1}(z^{2}) + zP_{k−1}(−z^{2}) ) · ( P_{k−1}(z^{−2}) + z^{−1}P_{k−1}(−z^{−2}) )

= P_{k−1}(z^{2})P_{k−1}(z^{−2}) + P_{k−1}(−z^{2})P_{k−1}(−z^{−2}) + {zP_{k−1}(−z^{2})P_{k−1}(z^{−2})+z^{−1}P_{k−1}(z^{2})P_{k−1}(−z^{−2})} .

The quantity in the braces only has odd powers, so its constant term vanishes. Hence

M_{1}(k) = CT [ P_{k}(z)P_{k}(z^{−1}) ] = CT [ P_{k−1}(z^{2})P_{k−1}(z^{−2}) ] + CT [ P_{k−1}(−z^{2})P_{k−1}(−z^{−2}) ] .

Replacing z^{2} by z in the first term on the right, and −z^{2} by z in the second term, does not change the constant term, hence, we have the linear recurrence equation with constant coefficients

M_{1}(k) = 2M_{1}(k − 1) ,

with the obvious initial condition M_{1}(0) = 1, that implies the explicit expression M_{1}(k) = 2^{k}. Equivalently, the generating function R_{1}(t) is given by

R_{1}(t) = \frac{1}{1 − 2t} .

Let’s move on to find an explicit formula for M_{2}(k) and/or R_{2}(t). That was already done by smart human John Littlewood ([L]) but let’s do it again.

Once again, let’s use the defining recurrence for the Rudin-Shapiro polynomials, but let’s abbreviate

a(k)(z) = P_{k}(z) , b(k)(z) = P_{k}(−z) , A(k)(z) = P_{k}(z^{−1}) , B(k)(z) = P_{k}(−z^{−1}) .

We have

P_{k}(z)^{2}P_{k}(z^{−1})^{2} = ( P_{k−1}(z^{2}) + zP_{k−1}(−z^{2}) )^{2} · ( P_{k−1}(z^{−2}) + z^{−1}P_{k−1}(−z^{−2}) )^{2} .

Expanding, discarding odd terms, replacing z^{2} by z, and using trivial symmetries due to the fact that the functional CT is preserved under the dihedral group {z → z, z → −z, z → z^{−1}, z → −z^{−1}}, we get that

CT [ a(k)^{2}A(k)^{2}] = 2CT [a(k−1)^{2}A(k−1)^{2} ] −2CT [ za(k−1)^{2}B(k−1)^{2}]+4CT [a(k−1)A(k−1)b(k−1)B(k−1) ] .

The first term is an old friend, our quantity of interest with k replaced by k − 1, but the other two are newcomers. So we do the same treatment to them. They in turn, may (and often do) introduce new quantities, but if all goes well, there would only be finitely many sequences, and we would get a finite system of first-order linear recurrences. This indeed happens, and one gets, for the generating functions of the encountered sequences, a system of six equations with six unknowns, and in particular, we get (in a split second, of course, we let Maple do it) that our desired object, the generating function of the sequence CT [a(k)^{2}A(k)^{2}], alias, R_{2}(t), is given by:

R_{2}(t) = \frac{4 t + 1}{(1 + 2 t) (1 − 4 t)} = \frac{4}{3} \frac{1}{1 − 4t} − \frac{1}{3} \frac{1}{1 + 2 t} .

By extracting the coefficient of t^{k}, we even get a nice explicit expression for M_{2}(k), already known to Littlewood

M_{2}(k) = \frac{4}{3}4^{k} − \frac{1}{3}(−2)^{k} .

This can be done for any monomial

z^{α_{0}}a(k)^{α_{1}}A(k)^{α_{2}}b(k)^{α_{3}}B(k)^{α_{4}} .

Define the sequence

E[α_{0}, α_{1}, α_{2}, α_{3}, α_{4}](k) := CT [ z^{α_{0}}a(k)^{α_{1}}A(k)^{α_{2}}b(k)^{α_{3}}B(k)^{α_{4}} ] .

Replacing a(k), A(k), b(k), B(k) by their expressions in terms of z, a(k−1), A(k−1), b(k−1), B(k−1), expanding, discarding odd terms, replacing z^{2} by z, and replacing each monomial by its canonical form, implied by the above-mentioned action of the dihedral group that preserves CT, we can express, each such E[.], in terms of other E[.]’s evaluated at k − 1. It is (presumably, they may some issues about powers of z) possible to show (and it has been done by Doche and Habsieger [DH], using a different approach) that this process terminates and eventually we will not get any new sequences, leaving us with a finite system of linear equations for the corresponding generating functions, that can be automatically solved, and lead to an expression in terms of a rational function, since we get a first-order system

F(t) = v + tAF(t) ,

(where F(t) is the vector of generating functions whose first component is our desired one), for some matrix A, of integers that the computer finds automatically, and our object of desire is the first component of F(t) = (I − tA)^{−1}v.

While it is painful for a human to do this, a computer does not mind, and the Maple package

HaroldSilentShapiro.txt

accompanying this article does it for any desired monomial in z, P_{k}(z), P_{k}(−z), P_{k}(z^{−1}), P_{k}(−z^{−1}). See the output files accompanying this article, that may be viewed from the front of this article

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/hss.html .

Unlike the beautiful approach of Doche and Habsieger, that uses clever human pre-processing to establish an algorithm, that was then hard-programmed by hand, our approach is naive “dynamical programming”, where we don’t make any a priori human analysis, and let the computer introduce new quantities as needed. To guarantee that it halts, we input a parameter, that we call K, and if the size of the system exceeds K it returns FAIL, leaving us the option to forget about it, or try again with a larger K.

Higher moments and Saffari’s Conjecture

Now that we have reduced, for any specific positive integer n, the computation of the generating function of the sequence of moments M_{n}(k), that we call R_{n}(t), to a routine calculation, we can ask our beloved computer to crank-out as many of them as it can output in a reasonable amount of time. According to the output file

http://www.math.rutgers.edu/~zeilberg/tokhniot/oHaroldSilentShapiro1.txt ,

we get

R_{1}(t) = \frac{1}{1 − 2t} ,

R_{2}(t) = \frac{1 + 4 t}{(1 + 2 t) (1 − 4 t)} ,

[both of which were already given above],

R_{3}(t) = \frac{1 + 16 t}{(1 + 4 t) (1 − 8 t)} ,

R_{4}(t) =

-(90194313216*t**11-15300820992*t**10-1979711488*t**9-292552704*t**8- 22216704*t**7+10649600*t**6-1024*t**5-144384*t**4+7008*t**3+664*t**2-54*t-1)/ ((8*t+1)* (16*t-1)*(1409286144*t**10-264241152*t**9-25690112*t**8-4128768*t**7-311296*t**6+ 170496*t**5-2624*t**4-2208*t**3+148*t**2+8*t-1)),

R_{5}(t) =

-(369435906932736*t**11-32160715112448*t**10-2001454759936*t**9-145223581696*t**8-4454350848*t**7+1392508928*t**6-5865472*t**5-4599808*t**4 +123648*t**3+4768*t**2-220*t-1)/ ((1+16*t)*(32*t-1)*(1443109011456*t**10-135291469824*t**9 -6576668672*t**8-528482304*t**7 -19922944*t**6+5455872*t**5-41984*t**4-17664*t**3+592*t**2+16*t-1)) .

To see R_{k}(t) for 6 ≤ k ≤ 10, look at the above-mentioned output file. Of course, one can easily go further. Note that these have already been computed in [DH] (but their output is not easily accessible to the casual reader).

By looking at the smallest root of the denominator of R_{k}(t) and computing the residue, one confirms for small (and not so small!) values of k (and one can easily go much further), the following conjecture of Bahman Saffari, as already done in [DH] (for small k).

Saffari’s Conjecture: For every positive integer n, as k → ∞, the following asymptotic formula holds.

M_{n}(k) ∼ \frac{2^{n}}{n + 1} · (2^{n})^{k} .

Saffari never published his conjecture, and it is mentioned as “private communication” in [DH].

Sketch of a proof of Saffari’s Conjecture

While for each numeric n, one can get an explicit expression, in symbolic t, for R_{n}(t), these get more and more complicated as n gets larger, and there is (probably) no hope to get an explicit expression, in symbolic n, for R_{n}(t), from which one can deduce that the smallest root (in absolute value) of the denominator is 2^{−n} and the residue is \frac{2^{n}}{n+1}.

But one can prove rigorously Saffari’s conjectured asymptotic formula as follows.

Let n be a general (symbolic) positive integer. Recall that we are interested in the sequence

M_{n}(k) := CT [P_{k}(z)^{n}P_{k}(z^{−1})^{n}] ,

that we abbreviate

CT [ a^{n}A^{n}] ,

under the convention

a = P_{k}(z) , b = P_{k}(−z) , A = P_{k}(z^{−1}) , B = P_{k}(−z^{−1}) .

To get a scheme we use the rewriting rules, implied by the defining recurrence

a → a + zb , b → a − zb , A → A + z^{−1}B , B → A − z^{−1}B ,

where the discrete argument on the left is k and on the right k − 1, and the continuous argument on the left is z and on the right is z^{2}.

Using the binomial theorem, we have

a^{n}A^{n} → (a + zb)^{n}(A + z^{−1}B)^{n} = (∑_{i=0}^{n} \binom{n}{i}a^{i}(zb)^{n−i}) (∑_{j=0}^{n} \binom{n}{j}A^{j}(z^{−1}B)^{n−j})

= ∑_{i=0}^{n} ∑_{j=0}^{n} \binom{n}{i}\binom{n}{j}a^{i}b^{n−i}A^{j}B^{n−j}z^{j−i}

= ∑_{i=0}^{n} \binom{n}{i}^{2}(aA)^{i}(bB)^{n−i} + SmallChange ,

where SmallChange is a linear combination of unimportant monomials and we define an important monomial (in a, A, b, B, z) to be any member of the set of monomials

{(aA)^{m}(bB)^{n−m} | 0 ≤ m ≤ n} .

Let’s try to find the “going down” evolution-step for the other important monomials.

We have

(aA)^{m}(bB)^{n−m} → (a + zb)^{m}(A + z^{−1}B)^{m}(a − zb)^{n−m}(A − z^{−1}B)^{n−m}

= (∑_{i_{1}=0}^{m} \binom{m}{i_{1}}a^{i_{1}}(zb)^{m−i_{1}}) (∑_{i_{2}=0}^{m} \binom{m}{i_{2}}A^{i_{2}}(z^{−1}B)^{m−i_{2}}) ·

( ∑_{i_{3}=0}^{n−m} \binom{n − m}{i_{3}}a^{i_{3}}(−zb)^{n−m−i_{3}}) ( ∑_{i_{4}=0}^{n−m} \binom{n − m}{i_{4}}A^{i_{4}}(−z^{−1}B)^{n−m−i_{4}})

= ∑_{i_{1}=0}^{m} ∑_{i_{2}=0}^{m} ∑_{i_{3}=0}^{n−m} ∑_{i_{4}=0}^{n−m} \binom{m}{i_{1}}\binom{m}{i_{2}}\binom{n − m}{i_{3}}\binom{n − m}{i_{4}}(−1)^{i_{3}+i_{4}}a^{i_{1}+i_{3}}A^{i_{2}+i_{4}}b^{n−i_{1}−i_{3}}B^{n−i_{2}−i_{4}}z^{i_{2}−i_{1}+i_{4}−i_{3}} .

The coefficient of a typical important monomial, (aA)^{r}(bB)^{n−r} (0 ≤ r ≤ n) in the above quadruple sum is

∑_{i_{1}+i_{3}=r , i_{2}+i_{4}=r}(−1)^{i_{3}+i_{4}} \binom{m}{i_{1}}\binom{m}{i_{2}}\binom{n − m}{i_{3}}\binom{n − m}{i_{4}}

= ∑_{i_{1}=0}^{r} ∑_{i_{2}=0}^{r}(−1)^{i_{1}+i_{2}} \binom{m}{i_{1}}\binom{m}{i_{2}}\binom{n − m}{r − i_{1}}\binom{n − m}{r − i_{2}}

= (∑_{i_{1}=0}^{r}(−1)^{i_{1}} \binom{m}{i_{1}}\binom{n − m}{r − i_{1}}) (∑_{i_{2}=0}^{r}(−1)^{i_{2}} \binom{m}{i_{2}}\binom{n − m}{r − i_{2}})

= (∑_{i=0}^{r}(−1)^{i}\binom{m}{i}\binom{n − m}{r − i})^{2} .

This is an important quantity, so let’s give it a name

K_{n}(m, r) := (∑_{i=0}^{r}(−1)^{i}\binom{m}{i}\binom{n − m}{r − i})^{2} .

All the remaining monomials belong to SmallChange, and we have the general ‘evolution equation’

(aA)^{m}(bB)^{n−m} → ∑_{r=0}^{n} K_{n}(m, r)(aA)^{r}(bB)^{n−r} + SmallChange .

Assuming for now, that SmallChange is, asymptotically less than the “important monomials” (i.e. the rate of growth of a small-change sequence divided by an “important monomial” sequence is o(1)), let α_{n} be the largest eigenvalue of the n + 1 by n + 1 matrix K_{n} (whose (m, r) entry is K_{n}(m, r)), then for 0 ≤ m ≤ n

CT [(P_{k}(z)P_{k}(z^{−1}))^{m}(P_{k}(−z)P_{k}(−z^{−1}))^{n−m}] ∼ c_{m}(α_{n})^{k} ,

where (c_{0}, …, c_{n}) is an eigenvector corresponding to the largest eigenvalue, α_{n}.

We now need two elementary propositions that should be provable using the Wilf-Zeilberger algorithmic proof theory, (a suitable extension of [Z][KKZ] for the first, [WZ][AZ] for the second). They may be even provable by purely human means, but since we know for sure that they are both true, we do not bother.

Proposition 1. The characteristic polynomial, det(zI − K_{n}), of the (n + 1) × (n + 1) matrix K_{n} whose (m, r) entry is

K_{n}(m, r) := (∑_{i=0}^{r}(−1)^{i}\binom{m}{i}\binom{n − m}{r − i})^{2} ,

equals

det(zI − K_{n}) = z^{⌊(n+1)/2⌋} ∏_{j=0}^{⌊n/4⌋} ( z − 2^{n−4j}\binom{4j}{2j} ) ∏_{j=0}^{⌊(n−2)/4⌋} ( z + 2^{n−4j−2}\binom{4j + 2}{2j + 1} ) .

[To confirm this shaloshable determinant identity for n ≤ N, type, in the Maple package

HaroldSilentShapiro.txt, CheckCP(N); . For example, CheckCP(20); returns true in one

second, and CheckCP(40); returns true in 20 seconds.]

So the non-zero eigenvalues of the matrix K_{n} are

\{ 2^{n−4j}\binom{4j}{2j} ; 0 ≤ j ≤ ⌊n/4⌋ \} ⋃ \{ −2^{n−4j−2}\binom{4j + 2}{2j + 1} ; 0 ≤ j ≤ ⌊(n − 2)/4⌋ \} .

In particular, the largest eigenvalue (in absolute value) is indeed 2^{n}. We also need the following shaloshable binomial coefficients identity.

Proposition 2. The vector (c_{0}, …, c_{n}) defined by c_{r} = \binom{n}{r}^{−1} (0 ≤ r ≤ n) is an eigenvector of the matrix K_{n} corresponding to its largest eigenvalue 2^{n} (with multiplicity 1). In other words, for 0 ≤ m ≤ n

∑_{r=0}^{n} K_{n}(m, r)c_{r} = 2^{n}c_{m} .

[To confirm this shaloshable binomial coefficient identity for n ≤ N, type, in the Maple pack-

age HaroldSilentShapiro.txt, CheckEV(N); . For example, CheckEV(50); returns true in two

seconds, and CheckCP(100); returns true in 30 seconds.]

But an eigenvector is only determined up to a constant multiple. Let’s find it (modulo the Small Change hypothesis). We know that

CT [ (P_{k}(z)P_{k}(z^{−1}))^{m}(P_{k}(−z)P_{k}(−z^{−1}))^{n−m}] ∼ \frac{C}{\binom{n}{m}} · (2^{n})^{k} ,

for some constant C. To find it, we use the well-known, and easily proved identity (see [Wi])

P_{k}(z)P_{k}(z^{−1}) + P_{k}(−z)P_{k}(−z^{−1}) = 2^{k+1} .

Raising it to the n-th power, using the binomial theorem, and taking the constant term, we have

∑_{m=0}^{n} \binom{n}{m}CT [(P_{k}(z)P_{k}(z^{−1}))^{m}(P_{k}(−z)P_{k}(−z^{−1}))^{n−m}] = 2^{(k+1)n} .

Hence

∑_{m=0}^{n} \binom{n}{m}\frac{C}{\binom{n}{m}} · (2^{n})^{k} = 2^{(k+1)n} ,

that implies that

C = \frac{2^{n}}{n + 1} .

We just established

Proposition 3: Modulo the Small Change Hypothesis, for 0 ≤ m ≤ n < ∞

CT [(P_{k}(z)P_{k}(z^{−1}))^{m}(P_{k}(−z)P_{k}(−z^{−1}))^{n−m}] ∼ \frac{2^{n}}{(n + 1)\binom{n}{m}} · (2^{n})^{k} .

In particular, taking m = n, we get Saffari’s conjecture (for even moments)

M_{n}(k) = CT [P_{k}(z)^{n}P_{k}(z^{−1})^{n}] ∼ \frac{2^{n}}{n + 1} · (2^{n})^{k} .

Towards a Proof of the Small Change Hypothesis

It would have been great if the “children” of each unimportant monomial, in the evolution equation described above (implemented in procedure GD in our Maple package), would all be unimportant. Then we could have easily proved, by induction that, asymptotically, they are insignificant compared to the important monomials. It turns out that for most unimportant monomials, this is indeed the case, but there are a few, that we call false pretenders that do have important children.

It should not be hard to fully characterize these. In fact it turns out (empirically, for now) that for n even there are (n/2)^{2} − 1 of them, and for n odd there are (n^{2} − 1)/4. Then for those false pretenders one should be able to describe all their important children, and then prove that the leading terms of their contributions cancel out (using the inductive hypothesis, and Prop. 3).

This has been verified empirically up to n ≤ 16. See procedures Medio and MedioP in the Maple

package HaroldSilentShapiro.txt .

Hugh Montgomery’s Stronger Conjecture

In [M], Hugh Montgomery considered the more general sequences

M_{m,n}(k) := CT [ P_{k}(z)^{m}P_{k}(z^{−1})^{n}] .

He conjectured that, for m ≠ n ,

M_{m,n}(k) = o( 2^{(m+n)k/2} ) .

Once again, the generating function, for each specific m and n, is always a rational function, and our Maple package (procedure RS(m,n,t,K)) computes them, and procedure MamarH(N,K,t) prints out an article confirming Hugh Montgomery’s conjecture, as well as giving the generating functions for 1 ≤ m < n ≤ N. (K is a parameter that should be made large enough, say 1000).

To see the output for 1 ≤ m < n ≤ 7, go to:

http://www.math.rutgers.edu/~zeilberg/tokhniot/oHaroldSilentShapiro2.txt ,

that contains the explicit expressions for all these cases, and confirms Montgomery’s conjecture with a vengeance. Unlike the m = n case, the smallest root (alias the reciprocal of the largest eigenvalue) is not ‘nice’, and there are usually several roots with smallest absolute value, hence the sequences often oscillate. Nevertheless, Montgomery’s conjecture is true for all 1 ≤ m < n ≤ 7, and one could go much further.

Let’s Generalize!

The same approach works for any sequence of Laurent polynomials defined by a 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}) ,

with the initial condition P_{0}(z) = 1, where C_{1}(z), C_{2}(z), C_{3}(z), C_{4}(z) are Laurent polynomials of degree less than r and low-degree larger than −r, for any positive integer r larger than 1.

One always gets a finite scheme (disclaimer: we don’t have a rigorous proof, but we believe that such a proof exists, at any rate, it is true in all the cases that we tried out) and hence a rational generating function for the sequence

S[α_{0}, α_{1}, α_{2}, α_{3}, α_{4}](k) := CT [ z^{α_{0}}P_{k}(z)^{α_{1}}P_{k}(z^{−1})^{α_{2}}P_{k}(−z)^{α_{3}}P_{k}(−z^{−1})^{α_{4}}] ,

for any non-negative α_{0}, α_{1}, α_{2}, α_{3}, α_{4}. This is implemented in the Maple package ShapiroGeneral.txt also available from the webpage of this article, or directly from

http://www.math.rutgers.edu/~zeilberg/tokhniot/ShapiroGeneral.txt .

Let’s (not!) Generalize Even More!

The set {1, −1} is a multiplicative subgroup of the field of complex numbers. For any finite multiplicative subgroup G of the field of complex numbers, and any positive integer r larger than 1, the same approach should be able to handle sequences of polynomials given by a recurrence

P_{k}(z) = ∑_{g∈G} α_{g}(z)P_{k−1}(gz^{r}) + ∑_{g∈G} β_{g}(z)P_{k−1}(gz^{−r}) , P_{0}(z) = 1 ,

where α_{g}(z), β_{g}(z) are 2|G| given Laurent polynomials in z of degree < r and low-degree > −r.

This includes the case treated in [D], where G is a cyclotomic group.

We could go even further, with higher order recurrences (as opposed to only first order), several continuous variables (as opposed to only z), and, presumably, even several discrete variables (as opposed to only k), but enough is enough!

Added May 27, 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 posted out in

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/BradleyRodgersLetter.pdf.

Added June 7, 2016: Brad Rodgers’ beautiful paper, that also proves the more general Montgomery conjecture, mentioned above, is now available here: http://arxiv.org/abs/1606.01637.

References

[AZ] Moa Apagodu and Doron Zeilberger, Multi-Variable Zeilberger and Almkvist-Zeilberger Algorithms and the Sharpening of Wilf-Zeilberger Theory, Adv. Appl. Math. 37 (2006), 139-152.

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/multiZ.html .

[D] Christophe Doche, Even moments of generalized Rudin-Shapiro polynomials, Math. Comp. 74(2005), 1923-1935.

http://www.ams.org/mcom/2005-74-252/S0025-5718-05-01736-9/S0025-5718-05-01736-9.pdf

[viewed May 6, 2016].

[DH] Christophe Doche and Laurent Habsieger, Moments of the Rudin-Shapiro polynomials, Journal of Fourier Analysis and Applications 10 (2004), 497-505.

http://web.science.mq.edu.au/~doche/049.pdf [viewed May 6, 2016].

[KKZ] Christoph Koutschan, Manuel Kauers, and Doron Zeilberger, A Proof Of George Andrews’ and David Robbins’ q-TSPP Conjecture, Proceedings of the National Academy of Science, 108, no. 6 (Feb. 8, 2011), 2196-2199.

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qtsppRig.html .

[L] John E. Littlewood, “Some Problems in Real and Complex Analysis”, D.C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968.

[M] Hugh Montgomery, Problem, in: “Problem Session”, p. 3029, Mathematisches Forschungsinstitut Oberwolfach, Report No. 51/2013, DOI: 10.4171/OWR/2013/51, Analytic Number Theory, 20 October - 26 October 2013, 3029-3030.

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May 11, 2016].

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May 11, 2016].

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http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/ansatzII.html .

Shalosh B. Ekhad, c/o D. Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

zeilberg at math dot rutgers dot edu ; http://www.math.rutgers.edu/~zeilberg/ .

First Written: May 20, 2016 ; Previous version: June 7, 2016 .

This version: Aug. 23, 2016 [Thanks to the insightful and careful referee report.]