Abstract
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 \(p>3\). 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 \(p=3, 5\), 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 \(p=2, 3, 5, 7, 11, 13\) and for \(p>15\), we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large n.
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1 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
each has non-negative coefficients.
Conjecture 1.2
(Second Borwein conjecture) For the polynomials \(\alpha _n(q), \beta _n(q)\) and \(\gamma _n(q)\) defined by
each has non-negative coefficients.
Conjecture 1.3
(Third Borwein conjecture) For the polynomials \(\nu _n(q), \phi _n(q), \chi _n(q)\), \(\psi _n(q)\) and \(\omega _n(q)\) defined by
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]:
Let \(p\ge 3\) be a prime and \(s, n \in {\mathbb {N}}\). Consider the polynomial
For \(s=1\), Borwein [2] obtained an asymptotic estimate for \(\left\| T_{p,1,n}(q)\right\| _{|q|=1}=\sup _{|q|=1}|T_{p,1,n}(q)|\) 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
where \(T_{i,p,s,n}(q)\in {\mathbb {Z}}[q]\). Given a polynomial f(x), by \([x^{j}]f(x)\), we denote the coefficient of \(x^{j}\) in f(x). Let \(a, d\in {\mathbb {Z}}\). In what follows, assume \(p\mid a\) and let \(S_{a,d,j,p}\) denote the arithmetic progression
Put \(a=p\ell \) 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 \(\ell \) is an odd prime \(\le n+1\) and \(\ell \ne p\). As a result, when \(\ell \le c(n+1)\) with \(0<c<1\), he showed positivity (resp. negativity) of the sum in (1.4) when \(j=0\) (resp. \(j\ne 0\)) 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 \(\ell \). When \(\ell \gg (n+1)^2\) (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 \(\ell \ll n+1\) which ensures that the main term is bigger than the error term, thereby showing positivity or negativity of the sums.
For \((p,s,\ell ,j)=(3,1,n+1,0)\), Li [4] obtained an asymptotic formula for the sum in (1.4) using a new sieve technique discovered by himself and Wan [3]. If we denote by \(t_{i,3,1}=a_{i}\), then Li proved that
Theorem 1.1
(Li) For \(0\le j\le (n+1)\) we have
In particular, we have
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 \(\ell =n+1\) 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.
2 Notation, conventions and basic counting
Let \(n\in {\mathbb {N}}\) and \(p\ge 3\) be a prime. Set \(N_p=(n+1)p\) and \(D_p=\{1,2,\ldots , p-1, p+1,\ldots ,2p-1,\ldots , pn+1,\ldots , pn+p-1\}\). We define the following:
![](http://media.springernature.com/lw522/springer-static/image/art%3A10.1007%2Fs11139-020-00352-0/MediaObjects/11139_2020_352_Equ86_HTML.png)
It is now apparent that
We note that in the case \(s=1\), \({\mathscr {C}}_{e,p,1}(j,n)\) (respectively \({\mathscr {C}}_{o,p,1}(j,n)\)) counts the number of partitions of j into an even (respectively odd) number of distinct non-multiples of p.
As in [4], we shift the problem to that of counting the size of certain subsets of the group \(G={\mathbb {Z}}_{N_p}\). We note that \(G\setminus D_{p}\) is a subgroup of index p. Given \(0\le k_1, k_2,\ldots ,k_s \le |D_{p}|\) and \(0\le b < N_p\), define
![](http://media.springernature.com/lw546/springer-static/image/art%3A10.1007%2Fs11139-020-00352-0/MediaObjects/11139_2020_352_Equ87_HTML.png)
and set
From (1.4) and (2.1), we see that if \(b\equiv j\;(\hbox {mod}\;p)\) then the following are equivalent:
where \(B_{p,s}(b):=\lfloor (d_{p,s,n}-b)/N_p\rfloor \). 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. Let \((x)_k:=x(x-1)(x-2)\cdots (x-k+1)\) denote the falling factorial. Let \({\hat{G}}\) be the set of complex-valued linear characters of G. By \(\psi _{0}\), we denote the trivial character in \({\hat{G}}\). Let \(X_{p,k}=D_p^k\) and \({\overline{X}}_{p,k}\) denote the subset of all tuples in \(D_p^{k}\) with distinct coordinates.
3 Main results
Our main results are below.
Theorem 3.1
With \(M_{p,s,n}(b)\) defined as in (2.2) and \(b\in {\mathbb {Z}}_{N_p}\) we have
where
Theorem 3.2
For a fixed prime \(p\ge 3\) and \(b\in {\mathbb {Z}}_{N_p}\), define
Then for all \(n\ge n_{p,s,b}\) we have
For \(p=3\), Li’s theorem [4, p. 4, Prop. 1.6] shows that \(M_{3,1,n}(b) (=M_{3,n}(b))>0\) when \(b\equiv 0\;(\hbox {mod 3})\) for all \(n>0\). For \(p=3\), when \(b\not \equiv 0\;(\hbox {mod} 3)\), we have
Theorem 3.3
Let \(b\equiv 1, 2\;(mod \;3)\) with \(b\in {\mathbb {Z}}_{N_3}\). Then
In particular, \(M_{3,1,n}(b)<0\) for all \(n>0\).
Theorem 3.4
For \(p=3, s=2\) and \(b\in {\mathbb {Z}}_{N_3}\) we have
In particular, \(M_{3,2,n}(b)>0\) (resp. \(M_{3,2,n}(b)<0\)) when \(b\equiv 0 \pmod 3\) (resp. \(b\not \equiv 0 \pmod 3\)) for all \(n>0\).
Theorem 3.5
For \(p=5, s=1\) and \(b\in {\mathbb {Z}}_{N_5}\) we have
In particular, \(M_{5,1,n}(b)>0\) (resp. \(M_{5,1,n}(b)<0\)) when \(b\equiv 0 \pmod 5\) (resp. \(b\not \equiv 0 \pmod 5\)) for all \(n>0\).
In view of (1.2), we immediately deduce the following from Theorem 3.1:
Corollary 3.5.1
For \(p\ge 3\) and \(b\in {\mathbb {Z}}_{N_p}\), let \(b\equiv j\;(mod \;p)\). Then we have
where \(\varSigma _{p,s,n,b}\) is as in Theorem 3.1 and \(B_{p,s}(b)=\lfloor (d_{p,s,n}-b)/N_p\rfloor .\)
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:
Corollary 3.5.2
For \(p=3, s=1\) and a fixed \(b\in {\mathbb {Z}}_{N_3}\), let \(b=3u+j\) be such that \(j\equiv 1, 2\;(mod \;3)\). Then we have
Corollary 3.5.3
For \(p=3, s=2\) and a fixed \(b\in {\mathbb {Z}}_{N_3}\), let \(b=3u+j\) be such that \(j\equiv 0, 1, 2\;(mod \;3)\). Then we have
Corollary 3.5.4
For \(p=5, s=1\) and \(b\in {\mathbb {Z}}_{N_5}\), let \(b\equiv j\;(mod \;5)\). Then we have
where \(T_{0,5,1,n}(q)=\nu _n(q),\; T_{1,5,1,n}(q)=\phi _n(q), \;T_{2,5,1,n}(q)=\chi _n(q),\; T_{3,5,1,n}(q)=\psi _n(q),\; T_{4,5,1,n}(q)=\omega _n(q)\) are the polynomials in Conjecture 1.3.
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
4 Li–Wan sieve
The quantity \(M_{p,s,n}(k_1,k_2,\ldots ,k_s;b)\) is the number of certain type of subsets of \(D_p^s\). As in [4] we apply some elementary character theory to estimate it.
We note that
is the regular character of G. It is well known that \(\rho (g)=0\) for all \(g\in G\setminus \{0\}\), and that \(\rho (0)=|G|=N_p\). Given \(0< r \le |D_{p}|\), a character \(\psi \in {\hat{G}}\), and \({\bar{x}}=(x_{1},\ldots ,\ x_{r})\), we set
Let \(Y_{p,s}^{k_1,k_2,\ldots ,k_s}\) denote the Cartesian product \(\prod _{i=1}^s {\overline{X}}_{p,k_i}\). Then we have
In the right-hand side above we interchange the sums to get
For a \(Y \subset X_{p,k}\) and a character \(\psi \in {\hat{G}}\), set \(F_{\psi }(Y):=\sum _{{\bar{y}} \in Y} f_{\psi }({\bar{y}})\). We now have
We now estimate sums of the form \(F_{\psi }({\overline{X}}_{p,k})\). The symmetric group \(S_{k}\) acts naturally on \(X_{p,k}=D_p^{k}\). Let \(\tau \in S_{k}\) be a permutation whose cycle decomposition is
where \(a_i\ge 1, 1\le i\le s\). We define
In other words, \(X_{p,k}^{\tau }\) is the set of elements in \(X_{p,k}\) fixed under the action of \(\tau \). Let \(C_{k}\) be a set of conjugacy class representatives of \(S_{k}\). Let us denote by \(C(\tau )\) the number of elements conjugate to \(\tau \). Now for any \(\tau \in S_{k}\), we have \(\tau (X_{p,k})=X_{p,k}\). We note that for any pair \(\tau \), \(\tau '\) of conjugate permutations, and for any \(\psi \in {\hat{G}}\), we have \(F_{\psi }({\overline{X}}^\tau _{n,k})=F_{\psi }({\overline{X}}^{\tau '}_{n,k_i})\). That is, according to the definitions in [3], \(X_{p,k}\) is symmetric and \(f_\psi \) is normal on X. Thus we have the following result which is essentially [3, Proposition 2.8].
Proposition 4.1
We have
4.1 Some useful lemmas
The following lemma exhibits the relationship between \(F_{\psi }({\overline{X}}^\tau _{p,k})\) and the cycle structure of \(\tau \).
Lemma 4.2
Let \(\tau \in C_{k}\) be the representative whose cyclic structure is associated with the partition \((1^{c_{1}},2^{c_{2}},\ldots k^{c_{k}})\) of k. Then we have \(F_{\psi }(X^{\tau }_{p,k})=\prod _{i=1}^{k}(\sum _{a \in D_p}\psi ^{i}(a))^{c_{i}}\).
Proof
Recall that
\(\square \)
Given \(\chi \in {\hat{G}}\) define
Let \(N(c_{1},c_{2},\ldots c_{k})\) denote the number of elements of \(S_{k}\) of cycle type \((c_{1},c_{2},\ldots c_{k})\). It is well known (see, for example, [5]) that
Then
Lemma 4.3
We have
Proof
To prove this lemma, we first note that \({{\,\mathrm{sgn}\,}}(\tau )=(-1)^{k-\sum _{i}c_i}\). Also the cyclic structure for every \(\tau \in C_k\) can be associated to a partition of k of the form \((1^{c_1}, 2^{c_2},\ldots ,k^{c_k})\). Hence the right-hand sum in Proposition 4.1 runs over all such partitions of k. Noting that the conjugate permutations have same cycle type, and there are exactly \(N(c_1,c_2,\ldots ,c_k)\) permutations with cycle type \((c_{1},c_{2},\ldots c_{k})\), we conclude, in view of Lemma 4.2, that
\(\square \)
Define the following polynomial in k variables:
From Lemma 4.3 and (4.5) we immediately see that
Corollary 4.3.1
We have
where for \(\chi \in {\hat{G}}\), \(s_{D_p}(\chi )\) is as in (4.3).
Thus, it only remains to evaluate the sums \(s_{D_p}(\chi )\) for \(\chi =\psi ^i, i=1, 2,\ldots , k\), and we do this next. Let \(o(\chi )\) denotes the order of the character \(\chi \). Then
Lemma 4.4
Let
and
Then
-
(1)
if \(p \not \mid o(\psi )\), \(F_{\psi }({\overline{X}}_{p,k})=(-1)^{k}Z_{k}(\delta _{1}^{\psi }(1),\ldots ,\ \delta _{1}^{\psi }(k))\), and
-
(2)
if \(p \mid o(\psi )\), \(F_{\psi }({\overline{X}}_{p,k})=(-1)^{k}Z_{k}(\delta _{2}^{\psi }(1),\ldots ,\ \delta _{2}^{\psi }(k))\).
Proof
First, observe that \(G\setminus D_p\) is a subgroup of index p. Hence from elementary character theory, we can deduce that
-
A.
if \(o(\psi )\ne 1,p\), we have \(s_{D_p}(\psi )=s_{G}(\psi )-s_{G\setminus D_p}(\psi )=0\),
-
B.
if \(o(\psi )=1\), we have \(s_{D_p}(\psi )=|D_p|=(p-1)N_p/p\), and
-
C.
if \(o(\psi )=p\), we have \(s_{D_p}(\psi )=-s_{G\setminus D_p}(\psi )=-|G\setminus D_p|=-N_p/p\).
In order to estimate \(F_{\psi }({\overline{X}}_{p,k})\), we need to consider the following two cases:
Case I: \(p \not \mid o(\psi )\). In this case, for all i we have \(p \not \mid o(\psi ^{i})\) since \(o(\psi ^i)=o(\psi )/(o(\psi ),i)\). Thus from (A) and (B) we see that
which implies (1) in view of Corollary 4.3.1 and the definition of \(\delta ^\psi _1(i)\).
Case II: \(p\mid o(\psi )\). Here we have the following from (A), (B) and (C):
which implies (2) in view of Corollary 4.3.1 and the definition of \(\delta ^\psi _2(i)\). \(\square \)
4.2 Some combinatorial functions and estimates
We now evaluate \(Z_{k}\left( \delta _{1}^{\psi }(1),\ldots , \delta _{1}^{\psi }(k)\right) \) and
\(Z_{k}(\delta _{1}^{\psi }(1),\ldots ,\ \delta _{1}^{\psi }(k))\). From (4.4) and (4.5) we immediately deduce the following:
Lemma 4.5
(Exponential generating function) We have
The next result follows by substituting special values for the variables \(t_1, t_2, \ldots \) in Lemma 4.5.
Corollary 4.5.1
We have
-
(1)
if \(t_i=a\) iff \(d\mid i\) and \(t_i=0\) iff \(d\not \mid i\), then
$$\begin{aligned} Z_k\left( \underbrace{0,\ldots ,0}_{d-1},a,\underbrace{0,\ldots ,0}_{d-1},a,\ldots \right) =\left[ \dfrac{u^k}{k!}\right] \dfrac{1}{(1-u^d)^{a/d}}. \end{aligned}$$ -
(2)
if \(t_i=a\) iff \(d\mid i\) and \(p\cdot d\not \mid i\); if \(t_i=b\) iff \(p\cdot d\mid i\); and if \(t_i=0\) iff \(d\not \mid i\), then
$$\begin{aligned}&Z_k\left( \overbrace{\underbrace{0,\ldots ,0}_{d-1},a,\underbrace{0,\ldots ,0}_{d-1},a,\underbrace{0,\ldots ,0}_{d-1}}^{p\cdot d-1},b,\ldots \right) \\&\quad =\left[ \dfrac{u^k}{k!}\right] \dfrac{1}{(1-u^d)^{a/d}(1-u^{pd})^{\frac{b-a}{pd}}}. \end{aligned}$$
Proof
The proof of this corollary is similar to the case for \(p=3\) in [4, p. 7, Lemma 2.3]. \(\square \)
From Lemma 4.4 and Corollary 4.5.1 we obtain
Lemma 4.6
We have
-
(1)
if \(p\not \mid o(\psi )\),
$$\begin{aligned} F_{\psi }({\overline{X}}_{p,k})=(-1)^k\left[ \dfrac{u^k}{k!}\right] (1-u^{o(\psi )})^{\frac{(p-1)N_p}{po(\psi )}} \end{aligned}$$ -
(2)
if \(p|o(\psi )\),
$$\begin{aligned} F_{\psi }({\overline{X}}_{p,k})=(-1)^k\left[ \dfrac{u^k}{k!}\right] \dfrac{(1-u^{o(\psi )})^{\frac{N_p}{o(\psi )}}}{(1-u^{o(\psi )/p})^{\frac{N_p}{o(\psi )}}}. \end{aligned}$$
5 Proofs of the main results
Proof of Theorem 3.1
From (4.2) we have
where
Using Lemma 4.6, we see that
Recall that
Using the well-known fact
we see that
Thus (5.1), (5.4) and (5.5) yield
Given a character \(\psi \) of order p, there is a unique \(x \in \{1,\ldots ,p-1\}\) such that for all \(y \in {\mathbb {Z}}_{N_p}\), we have \(\psi (y)=e^{2\pi i xy/p}\). Now
Case I: If \(p \mid b\). Then
Case II: If \(p \not \mid b\). Then, as x runs over elements in \(Z_p^{\times }\), so does bx and we get
So from (5.3) we have
Noting that the sum
since it is the sum of all coefficients of the multinomial expansion of \((1+u+u^2+\cdots +u^{p-1})^{N_p/p}\), we obtain the following from (5.7), (5.8) and (5.9):
Next, we estimate \(P_{k_1,k_2,\ldots ,k_s}\) and \(R_{k_1,k_2,\ldots ,k_s}\). Consider
Finally, we consider
where the last step is obtained by noting that \(\sum _{k=0}^{|D_{p}|}[u^{k}](1-u^{j})^{\frac{(p-1)N_p}{pj}}\) is the sum of all coefficients of \((1-u^{j})^{\frac{(p-1)N_p}{pj}}\) which is zero.
Hence (5.6), (5.11), (5.12) and (5.13) yield
which yields the theorem. \(\square \)
Proof of Theorem 3.2
To prove this theorem, we need to consider two cases.
Case I: \(p\mid b\). In this case, from Theorem 3.1 we have
Thus the main term in Theorem 3.1 dominates the error provided
It is now clear that for all \(n\ge \inf \{n\in {\mathbb {N}}: (p-1)p^{s(n+1)/2-1}>n+1\}\), \(M_{p,s,b}(b)>0\) when \(p\mid b\).
Case II: \(p\not \mid b\). In this case, from Theorem 3.1 we have
Thus the absolute value of the main term in Theorem 3.1 dominates the error provided
It is now clear that for all \(n\ge \inf \{n\in {\mathbb {N}}: p^{s(n+1)/2-1}>n+1\}\), \(M_{p,s,n}(b)<0\) when \(p\not \mid b\). This proves the theorem. \(\square \)
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 \(n_{3,1,b}\in {\mathbb {N}}\) for which
holds true is \(n_{3,1,b}=4\). Thus for all \(n\ge 4\) 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. \(\square \)
Proof of Theorem 3.4
The first part of this theorem follows directly from Theorem 3.1 by choosing \(p=3\) and \(s=2\).
For the other part, we use Theorem 3.2. Thus in the case \(b\equiv 0 \pmod 3\) we have
for all \(n\in {\mathbb {N}}\). Hence \(M_{3,2,n}>0\) for all \(n\in {\mathbb {N}}\). In the case \(b\not \equiv 0 \pmod 3\) we have
holds true for all \(n\in {\mathbb {N}}\). Hence \(M_{3,2,n}<0\) for all \(n\in {\mathbb {N}}\). \(\square \)
Proof of Theorem 3.5
The first part of this theorem follows directly from Theorem 3.1 by choosing \(p=5\) and \(s=1\).
For the other part, we use Theorem 3.2. Thus in the case \(b\equiv 0 \pmod 5\) we have
for all \(n\in {\mathbb {N}}\). Hence \(M_{5,1,n}>0\) for all \(n\in {\mathbb {N}}\). In the case \(b\not \equiv 0 \pmod 5\) we have
holds true for all \(n\ge 3\). By direct computation one checks that \(M_{5,1,n}<0\) for all \(n<3\). Hence \(M_{5,1,n}<0\) for all \(n\in {\mathbb {N}}\). \(\square \)
Proof of Theorem 3.6
Using Cauchy’s formula we see that
On the other hand we have
Since \(d_{p,s,n}\le sp^2n^2\), (5.16) and (5.17) imply
Thus the theorem follows if we show that
We note that
From [2, Theorem 1, p. 229], we have
Now the estimate (5.19) and thus the theorem follow from (5.18), (5.20) and (5.21). \(\square \)
Proof of Theorem 3.7
We have
which implies since \(d_{p,s,n}\le sp^2n^2\) that
From [2, Theorem 2, p. 229] we have
Since
\(\square \)
References
Andrews, G.: On a conjecture of Peter Borwein. J. Symbol. Comput. 20, 487–501 (1995)
Borwein, P.: Some restricted partition functions. J. Number Theory 45, 228–240 (1993)
Li, J., Wan, D.: A new sieve for distinct coordinate counting. Sci. China Math. (Springer) 53(9), 2351–2362 (2010)
Li, J.: On the Borwein conjecture. Int. J. Number Theory 16(5), 1053–1066 (2020)
Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge University Press, Cambridge (1997)
Wang, C.: An analytic proof of the Borwein conjecture. arXiv: 1901.10886 (2019)
Zaharescu, A.: Borwein’s conjecture on average over arithmetic progression. Ramanujan J. 11, 95–102 (2006)
Acknowledgements
The authors thank George Andrews, Peter Paule, Qing Xiang and Cai-Heng Li for their feedback. We also thank the anonymous referee for valuable suggestions and feedback.
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Goswami, A., Pantangi, V.R.T. On sums of coefficients of polynomials related to the Borwein conjectures. Ramanujan J 57, 369–387 (2022). https://doi.org/10.1007/s11139-020-00352-0
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DOI: https://doi.org/10.1007/s11139-020-00352-0