On sums of coefficients of polynomials related to the Borwein conjectures

Recently, Li (Int J Number Theory, 2020) obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan (Sci China Math, 2010). In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>3$$\end{document}p>3. In this work, we extend Li’s method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in the special cases \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=3, 5$$\end{document}p=3,5, the signs of these sums are consistent with the three famous Borwein conjectures. Similar sums have been studied earlier by Zaharescu (Ramanujan J, 2006) using a completely different method. We also improve on the error terms in the asymptotic formula for Li and Zaharescu. Using a recent result of Borwein (JNT 1993), we also obtain an asymptotic estimate for the maximum of the absolute value of these coefficients for primes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2, 3, 5, 7, 11, 13$$\end{document}p=2,3,5,7,11,13 and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>15$$\end{document}p>15, we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large n.


Introduction
In 1990, Peter Borwein (see [1]) empirically discovered quite a number of mysteries involving sign patterns of coefficients of certain polynomials. The most easily stated are the following: Conjecture 1.1 (First Borwein conjecture) For the polynomials A n (q), B n (q) and C n (q) defined by n j=1 each has non-negative coefficients.
Recently, Wang [6] gave an analytic proof of the First Borwein conjecture using saddle point method. His proof, besides other things, relied on a formula of Andrews [1,Theorem 4.1] and the following recursive relations [1, Theorem 3.1]: A n (q) = (1 + q 2n−1 )A n−1 (q) + q n B n−1 (q) + q n C n−1 (q), B n (q) = q n−1 A n−1 (q) + (1 + q 2n−1 )B n−1 (q) − q n C n−1 (q), C n (q) = q n−1 A n−1 (q) + q n−1 B n−1 (q) − (1 + q 2n−1 )C n−1 (q). For s = 1, Borwein [2] obtained an asymptotic estimate for T p,1,n (q) |q|=1 = sup |q|=1 |T p,1,n (q)| when p = 2, 3, 5, 7, 11 and 13. However for p > 15, he obtained an asymptotic lower bound for this quantity. It is clear that d p,s,n := deg T p,s,n = p( p − 1)s(n + 1) 2 /2. Define the coefficients t i, p,s by T p,s,n (q) := d p,s,n i=0 t i, p,s q i = T 0, p,s,n (q p ) + qT 1, p,s,n (q p ) + · · · + q p−1 T p−1, p,s,n (q p ), (1.2) where T i, p,s,n (q) ∈ Z [q]. Given a polynomial f (x), by [ In what follows, assume p | a and let S a,d, j, p denote the arithmetic progression Put a = p and consider the following finite sum of coefficients over S a,d, j, p : In [7, p. 98, Theorem 1], Zaharescu obtained an asymptotic formula for the sum in (1.4) when is an odd prime ≤ n + 1 and = p. As a result, when ≤ c(n + 1) with 0 < c < 1, he showed positivity (resp. negativity) of the sum in (1.4) when j = 0 (resp. j = 0) for large n. As Zaharescu points out in his paper, it is interesting to obtain positivity (or negativity) of the above sum for larger values of . When (n + 1) 2 (with implied constant larger than 1), one can isolate each individual terms in the sum (1.4). We note here that the main disadvantage of his asymptotic formula is the error term, which is large. This forces him to choose a n + 1 which ensures that the main term is bigger than the error term, thereby showing positivity or negativity of the sums.
Indeed, the error term in Li's asymptotic formula [4, p. 4, Theorem 1.5] is much better than Zaharescu's which enabled him to prove the positivity of the sum in Theorem 1.1. The purpose of this paper is to extend Li's results by obtaining asymptotic formula for the sums in (1.4) in the case = n + 1 for all p, s, j. As a consequence, we obtain positivity (or negativity) of the sums in (1.4) for large n. Thus, for p = 3, 5, we obtain asymptotic formula for the partial sums of coefficients involving polynomials in Conjectures 1.1-1.3. This in turn shows that the sums are positive (or negative) for all n > 0 (see Corollaries 3.5.2-3.5.4). We also improve on the error terms in Li's and Zaharescu's asymptotic formula. Using a recent result of Borwein [2], we also obtain an asymptotic estimate for the maximum absolute coefficients of T p,s,n (q) only in the case p = 2, 3, 5, 7, 11, 13; however for p > 15 we obtain an asymptotic lower bound for the maximum absolute coefficients.
This paper is organized as follows. In Sect. 2 we introduce a few notations, conventions and do some basic counting. In Sect. 3 we state our main results. In Sect. 4 we recall Li and Wan's [3] sieving principle and also establish a few basic results. Finally in Sect. 5 we obtain the proofs of our main results.

Notation, conventions and basic counting
Let n ∈ N and p ≥ 3 be a prime. Set N p = (n + 1) p and D p = {1, 2, . . . , p − 1, p + 1, . . . , 2 p − 1, . . . , pn + 1, . . . , pn + p − 1}. We define the following: It is now apparent that We note that in the case s = 1, C e, p,1 ( j, n) (respectively C o, p,1 ( j, n)) counts the number of partitions of j into an even (respectively odd) number of distinct nonmultiples of p. As in [4], we shift the problem to that of counting the size of certain subsets of the group G = Z N p . We note that G \ D p is a subgroup of index p. Given 0 ≤ k 1 , k 2 , . . . , k s ≤ |D p | and 0 ≤ b < N p , define M p,s,n (k 1 , k 2 , . . . , k s ; b) For ease of notation, we will mostly use the second sum in (2.2) for M p,s,n (b). We next introduce a few more notations.
) denote the falling factorial. LetĜ be the set of complex-valued linear characters of G. By ψ 0 , we denote the trivial character inĜ. Let X p,k = D k p and X p,k denote the subset of all tuples in D k p with distinct coordinates.

Main results
Our main results are below.

Theorem 3.2 For a fixed prime p
Then for all n ≥ n p,s,b we have (1)).
In view of (1.2), we immediately deduce the following from Theorem 3.1: In particular, noting the fact that the polynomials T j, p,s,n (q) are the polynomials in the first three Borwein conjectures for suitable choices of j, p and s we have, in view of Theorems 3.3-3.5 the following: (1)). (1)).
Theorem 3.6 Let p = 2, 3, 5, 7, 11, 13 and n be sufficiently large. Then we have Theorem 3.7 Let p > 15 and n be sufficiently large. Then we have

Li-Wan sieve
The quantity M p,s,n (k 1 , k 2 , . . . , k s ; b) is the number of certain type of subsets of D s p . As in [4] we apply some elementary character theory to estimate it.
We note that ρ := ψ∈Ĝ ψ is the regular character of G. It is well known that ρ(g) = 0 for all g ∈ G \{0}, and that ρ(0) = |G| = N p . Given 0 < r ≤ |D p |, a character ψ ∈Ĝ, andx = (x 1 , . . . , x r ), we set denote the Cartesian product s i=1 X p,k i . Then we have In the right-hand side above we interchange the sums to get For a Y ⊂ X p,k and a character ψ ∈Ĝ, set F ψ (Y ) := ȳ∈Y f ψ (ȳ). We now have We now estimate sums of the form F ψ (X p,k ). The symmetric group S k acts naturally on X p,k = D k p . Let τ ∈ S k be a permutation whose cycle decomposition is τ = (i 1 i 2 · · · i a 1 )( j 1 j 2 · · · j a 2 ) · · · ( 1 2 · · · a s ), In other words, X τ p,k is the set of elements in X p,k fixed under the action of τ . Let C k be a set of conjugacy class representatives of S k . Let us denote by C(τ ) the number of elements conjugate to τ . Now for any τ ∈ S k , we have τ (X p,k ) = X p,k . We note that for any pair τ , τ of conjugate permutations, and for any ψ ∈Ĝ, we have That is, according to the definitions in [3], X p,k is symmetric and f ψ is normal on X . Thus we have the following result which is essentially [3, Proposition 2.8].

Some useful lemmas
The following lemma exhibits the relationship between F ψ (X τ p,k ) and the cycle structure of τ .

Lemma 4.2
Let τ ∈ C k be the representative whose cyclic structure is associated with the partition (1 c 1 , 2 Let N (c 1 , c 2 , . . . c k ) denote the number of elements of S k of cycle type (c 1 , c 2 , . . . c k ). It is well known (see, for example, [5]) that Then Lemma 4. 3 We have Proof To prove this lemma, we first note that sgn(τ ) = (−1) k− i c i . Also the cyclic structure for every τ ∈ C k can be associated to a partition of k of the form Define the following polynomial in k variables: From Lemma 4.3 and (4.5) we immediately see that where for χ ∈Ĝ, s D p (χ ) is as in (4.3).
Thus, it only remains to evaluate the sums s D p (χ ) for χ = ψ i , i = 1, 2, . . . , k, and we do this next. Let o(χ ) denotes the order of the character χ . Then

Lemma 4.4 Let
In order to estimate F ψ (X p,k ), we need to consider the following two cases:

i). Thus from (A) and (B) we see that
which implies (2)

Some combinatorial functions and estimates
We now evaluate Z k δ Corollary 4.5. 1 We have Proof The proof of this corollary is similar to the case for p = 3 in [4, p. 7, Lemma 2.3].
From Lemma 4.4 and Corollary 4.5.1 we obtain Lemma 4. 6 We have . where

Proofs of the main results
Using Lemma 4.6, we see that . Using the well-known fact we see that (5.9) Noting that the sum since it is the sum of all coefficients of the multinomial expansion of (1 + u + u 2 + · · · + u p−1 ) N p / p , we obtain the following from (5.7), (5.8) and (5.9): Σ p,s,n,b := Next, we estimate P k 1 ,k 2 ,...,k s and R k 1 ,k 2 ,...,k s . Consider It is now clear that for all n ≥ inf{n ∈ N : p s(n+1)/2−1 > n + 1}, M p,s,n (b) < 0 when p b. This proves the theorem.

Proof of Theorem 3.3
The first part of the theorem follows from Theorem 3.1 by choosing p = 3 and s = 1. For the other part, we use Theorem 3.2. Thus the smallest n 3,1,b ∈ N for which 3 (n+1)/2−1 > n + 1 holds true is n 3,1,b = 4. Thus for all n ≥ 4 we have M 3,1,n (b) < 0. Also by direct computation, one shows that M 3,1,n (b) < 0 for all n < 4. Indeed, using Wang's result [6], one immediately concludes that M 3,1,n (b) < 0 without any of the above analysis. holds true for all n ≥ 3. By direct computation one checks that M 5,1,n < 0 for all n < 3. Hence M 5,1,n < 0 for all n ∈ N.