Positive Weighted Partitions Generated by Double Series
We investigate some weighted integer partitions whose generating functions are double-series. We will establish closed formulas for these -double series and deduce that their coefficients are non-negative. This leads to inequalities among integer partitions.
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
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| Мова: | Англійська |
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Інститут математики НАН України
2025
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Positive Weighted Partitions Generated by Double Series. George E. Andrews and Mohamed El Bachraoui. SIGMA 21 (2025), 056, 12 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860298616842747904 |
|---|---|
| author | Andrews, George E. El Bachraoui, Mohamed |
| author_facet | Andrews, George E. El Bachraoui, Mohamed |
| citation_txt | Positive Weighted Partitions Generated by Double Series. George E. Andrews and Mohamed El Bachraoui. SIGMA 21 (2025), 056, 12 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We investigate some weighted integer partitions whose generating functions are double-series. We will establish closed formulas for these -double series and deduce that their coefficients are non-negative. This leads to inequalities among integer partitions.
|
| first_indexed | 2026-03-21T18:50:11Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 056, 12 pages
Positive Weighted Partitions Generated
by Double Series
George E. ANDREWS a and Mohamed EL BACHRAOUI b
a) The Pennsylvania State University, University Park, Pennsylvania 16802, USA
E-mail: andrews@math.psu.edu
b) United Arab Emirates University, PO Box 15551, Al-Ain, United Arab Emirates
E-mail: melbachraoui@uaeu.ac.ae
Received March 14, 2025, in final form July 04, 2025; Published online July 12, 2025
https://doi.org/10.3842/SIGMA.2025.056
Abstract. We investigate some weighted integer partitions whose generating functions are
double-series. We will establish closed formulas for these q-double series and deduce that
their coefficients are non-negative. This leads to inequalities among integer partitions.
Key words: partitions; q-series; positive q-series
2020 Mathematics Subject Classification: 11P81; 05A17; 11D09
In honor of Steve Milne’s 75th birthday
1 Introduction
Throughout, let q denote a complex number satisfying |q| < 1 and let m and n denote nonneg-
ative integers. We adopt the following standard notation from the theory of q-series [4, 15]
(a; q)0 = 1, v(a; q)n =
n−1∏
j=0
(
1− aqj
)
, (a; q)∞ =
∞∏
j=0
(
1− aqj
)
,
(a1, . . . , ak; q)n =
k∏
j=1
(aj ; q)n, and (a1, . . . , ak; q)∞ =
k∏
j=1
(aj ; q)∞.
We shall need the following basic facts
(a; q)n+m = (a; q)m(aqm; q)n and (a; q)∞ = (a; q)n(aq
n; q)∞. (1.1)
In this paper, we consider certain q-double series in one single variable which turn out to be
natural generating functions for weighted integer partitions.
Weighted integer partitions have been extensively studied in the past. A first systematic
investigation of identities for weighted partitions is due to Alladi [1, 2]. For other references on
weighted partitions and their applications, see, for instance, [14, 20].
A power series
∑
n≥0 anq
n is called positive, written
∑
n≥0 anq
n ⪰ 0, if an ≥ 0 for any
nonnegative integer n. Accordingly, we will write
∑
n≥0 anq
n ⪰
∑
n≥0 bnq
n to mean that∑
n≥0 anq
n −
∑
n≥0 bnq
n ⪰ 0. Positivity results for q-series have been intensively studied in
the past to some extent in connection with Borwein’s famous positivity conjecture [3]. For
This paper is a contribution to the Special Issue on Basic Hypergeometric Series Associated with Root
Systems and Applications in honor of Stephen C. Milne’s 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Milne.html
mailto:andrews@math.psu.edu
mailto:melbachraoui@uaeu.ac.ae
https://doi.org/10.3842/SIGMA.2025.056
https://www.emis.de/journals/SIGMA/Milne.html
2 G.E. Andrews and M. El Bachraoui
more on this, see, for instance, [11, 12, 19, 21]. Positivity results for alternating sums have also
received much attention in recent years, see for example [6, 7, 13, 16].
An important application of positivity results is the fact that positive series which are gen-
erating functions for weighted partitions give rise to inequalities of integer partitions. About
the interplay between positive q-series and inequalities of integer partitions, we refer the reader
to [5, 8, 9, 10, 18].
Our main goal in this work is to prove that certain q-double series are positive. As these
series turn out to be generating functions for weighted partitions, our results yield inequalities
of integer partitions.
The paper is organized as follows. In Section 2, we introduce our series through the partitions
they generate and we state our main results. In Section 3, we collect the lemmas needed to
prove the main theorems. Sections 4–6 are devoted to the proofs of the main results and their
corollaries.
2 Main results
Definition 2.1. For any positive integer N , let F1(N) be the number of partitions of N , where
if the partition has n ones then the largest part is 2n+ 2k + 1 for some k and all parts > 1 are
in the interval [2n + 2, 2n + 2k + 1], no even parts are repeated, and the partition is counted
with weight (−1)j , where j is the number of even parts. Then we have
∑
n≥0
F1(n)q
n =
∑
k,n≥0
(
q2n+2; q2
)
k(
q2n+3; q2
)
k
q2k+3n+1.
We now state our first main result.
Theorem 2.2. We have
∑
n≥0
F1(n)q
n =
1(
1− q2
) (q2; q2)2∞(
q; q2
)2
∞
−
(
1 + q2
)
(1− q)
(
1− q3
) .
With the help of Theorem 2.2, we will derive the following positivity result.
Corollary 2.3. There holds
∑
n≥0 F1(n)q
n ⪰ 0.
We now introduce our second integer partitions.
Definition 2.4. For any positive integer N , let F2(N) be the number of partitions of N such
that for each j = 0, 1, 2 satisfying 3 | 2k + j for some k, there are n+ j − 2 ones and (2k + j)/3
threes, the remaining parts lie in the set {2n + 2} ∪ (2n + 3, 2n + 2k + 3], no even parts are
repeated, and the partition is counted with weight (−1)j , where j is the number of even parts.
Then we have∑
n≥0
F2(n)q
n =
∑
k,n≥0
(
q2n+2; q2
)
k(
q2n+5; q2
)
k
q2k+n+2.
Theorem 2.5. We have
∑
n≥0
F2(n)q
n =
q
(
1− q3
)
(1− q)
(
1− q2
) (q2; q2)2∞(
q; q2
)2
∞
−
q
(
1 + q2
)
(1− q)2
.
Corollary 2.6. There holds
∑
n≥0 F2(n)q
n ⪰ 0.
Positive Weighted Partitions Generated by Double Series 3
We now deal with our third example of integer partitions.
Definition 2.7. For any positive integer N , let G(N) be the number of partitions of N , where
if the partition has 3n ones then the largest part is 2n+ 2k + 2 for some k and all parts are in
the interval [2n+ 1, 2n+ 2k + 2], no even parts are repeated, and the partition is counted with
weight (−1)j , where j is the number of even parts. Then we have
∑
n≥0
G(n)qn =
∑
k,n≥0
(
q2n+2; q2
)
k(
q2n+1; q2
)
k
q2k+5n+2.
Theorem 2.8. We have
∑
n≥0
G(n)qn =
q3
(1 + q)
(
1− q3
) (q2; q2)2∞(
q; q2
)2
∞
−
q2(1− q)
(
−1 + q3 + q4 + q5
)(
1− q3
)2(
1− q5
) .
Corollary 2.9. There holds
∑
n≥0G(n)q
n ⪰ 0.
Our proofs for Corollaries 2.3, 2.6 and 2.9 on positive weighted partitions are all analytic.
Obviously, each of these three corollaries is equivalent to an inequality of integer partitions. So,
it is natural to ask for injective proofs for these inequalities.
3 Preliminary lemmas
In this section, we collect several lemmas which we will need to prove our main results. To
simplify the presentation, we introduce the following sequences.
Definition 3.1. For any positive integers m and n, let
∑
n≥0
A(m,n)qn =
∑
n≥0
qmn
(
q2n+2; q2
)
∞
(
q2n+4; q2
)
∞(
q2n−1; q2
)
∞
(
q2n+1; q2
)
∞
,
∑
n≥0
A′(m,n)qn =
∑
n≥0
qm(n+1)
(
q2n+2; q2
)
∞
(
q2n+6; q2
)
∞(
q2n+1; q2
)
∞
(
q2n+3; q2
)
∞
,
∑
n≥0
B(m,n)qn =
∑
n≥0
qmn
(
q2n+2; q2
)
∞
(
q2n+4; q2
)
∞(
q2n−3; q2
)
∞
(
q2n+3; q2
)
∞
,
∑
n≥0
B′(m,n)qn =
∑
n≥0
qm(n+1)
(
q2n+2; q2
)
∞
(
q2n+6; q2
)
∞(
q2n−1; q2
)
∞
(
q2n+5; q2
)
∞
.
Lemma 3.2. We have
∑
n≥0
A(2, n)qn =
−q
(
1 + q2
)
(1− q)2
(
1− q3
) (3.1)
and for any positive integer m ≥ 2,
∑
n≥0
A(2m,n)qn =
−1
(1− q)2
∑
n≥0
qn+1
(
q2n+2; q2
)
m−1(
q2n+3; q2
)
m−2
. (3.2)
4 G.E. Andrews and M. El Bachraoui
Proof. Throughout we will use Heine’s transformations [15, Appendix III, equations (III.1)–
(III.2)]
2ϕ1
[
a, b
c
; q, z
]
=
(b, az; q)∞
(c, z; q)∞
2ϕ1
[
c/b, z
az
; q, b
]
, (3.3)
=
(c/b, bz; q)∞
(c, z; q)∞
2ϕ1
[
abz/c, b
bz
; q, c/b
]
, (3.4)
where
2ϕ1
[
a, b
c
; q, z
]
=
∑
n≥0
(a; q)n(b; q)n
(q; q)n(c; q)n
zn.
We will also use the q-binomial theorem [4, equation (2.2.1)],∑
n≥0
(a; q)n
(q; q)n
zn =
(az; q)∞
(z; q)∞
(3.5)
and Ramanujan’s 1ψ1 summation formula [15, equation (5.2.1)]
∞∑
n=−∞
(a; q)n
(b; q)n
zn =
(q, b/a, az, q/(az); q)∞
(b, q/a, z, b/(az); q)∞
. (3.6)
We start with (3.1). By (1.1) and (3.4), we obtain
∑
n≥0
A(2, n)q2n =
∑
n≥0
q2n
(
q2n+2, q2n+4; q2
)
∞(
q2n−1, q2n+1; q2
)
∞
=
(
q2, q4; q2
)
∞(
q−1, q; q2
)
∞
∑
n≥0
(
q−1, q; q2
)
n
q2n(
q2, q4; q2
)
n
=
(
q2, q4; q2
)
∞(
q−1, q; q2
)
∞
2ϕ1
[
q−1, q
q4
; q2, q2
]
=
(
q2, q4; q2
)
∞(
q−1, q; q2
)
∞
(
q3, q3; q2
)
∞(
q2, q4; q2
)
∞
2ϕ1
[
q−2, q
q3
; q2, q3
]
=
−q
(1− q)3
(
1−
q(1− q)
(
1− q2
)(
1− q2
)(
1− q3
)) =
−q
(
1 + q2
)
(1− q)2
(
1− q3
) .
Now assume that m is a positive integer such that m ≥ 2. We get by using (1.1) and (3.3)
∑
n≥0
A(2m,n)qn =
∑
n≥0
q2mn
(
q2n+2, q2n+4; q2
)
∞(
q2n−1, q2n+1; q2
)
∞
=
(
q2, q4; q2
)
∞(
q−1, q; q2
)
∞
∑
n≥0
(
q−1, q; q2
)
n
q2mn(
q2, q4; q2
)
n
=
(
q2, q4; q2
)
∞(
q−1, q; q2
)
∞
2ϕ1
[
q−1, q
q4
; q2, q2m
]
=
(
q2, q4; q2
)
∞(
q−1, q; q2
)
∞
(
q, q2m−1; q2
)
∞(
q4, q2m; q2
)
∞
2ϕ1
[
q3, q2m
q2m−1
; q2, q
]
=
(
q2; q2
)
∞(
q−1; q2
)
∞
(
q2m−1; q2
)
∞(
q2m; q2
)
∞
(
q3, q2m; q2
)
∞(
q2, q2m−1; q2
)
∞
∑
n≥0
qn
(
q2n+2, q2n+2m−1; q2
)
∞(
q2n+3, q2n+2m; q2
)
∞
=
−q
(1− q)2
∑
n≥0
qn
(
q2n+2; q2
)
m−1(
q2n+3; q2
)
m−2
=
−1
(1− q)2
∑
n≥1
qn
(
q2n; q2
)
m−1(
q2n+1; q2
)
m−2
,
which confirms (3.2). ■
Positive Weighted Partitions Generated by Double Series 5
Lemma 3.3. For any positive integer, we have∑
n≥0
A′(2m,n)qn =
1
1− q3
∑
n≥0
qn+2m
(
q2n+2; q2
)
m−1(
q2n+5; q2
)
m−1
(3.7)
=
1
1− q
∑
n≥0
q3n+2m
(
q2n+2; q2
)
m−1(
q2n+3; q2
)
m−1
. (3.8)
Proof. We start with (3.7). We have by (1.1)
∑
n≥0
A′(2m,n)qn =
∑
n≥0
q2m(n+1)
(
q2n+2; q2
)
∞
(
q2n+6; q2
)
∞(
q2n+1; q2
)
∞
(
q2n+3; q2
)
∞
= q2m
(
q2, q6; q2
)
∞(
q, q3; q2
)
∞
∑
n≥0
q2m
(
q, q3; q2
)
n(
q2, q6; q2
)
n
= q2m
(
q2, q6; q2
)
∞(
q, q3; q2
)
∞
2ϕ1
[
q, q3
q6
; q2, q2m
]
= q2m
(
q2, q6; q2
)
∞(
q, q3; q2
)
∞
(
q, q2m+3; q2
)
∞(
q6, q2m; q2
)
∞
2ϕ1
[
q5, q2m
q2m+3
; q2, q
]
=
q2m
1− q3
∑
n≥0
qn
(
q2n+2, q2m+2n+3; q2
)
∞(
q2n+5, q2m+2n; q2
)
∞
=
1
1− q3
∑
n≥0
q2m+n
(
q2n+2; q2
)
m−1(
q2n+5; q2
)
m−1
,
where in the fourth step we applied (3.3) with (a, b, c, z) =
(
q3, q, q6, q2m
)
. This proves the
desired result.
As for (3.8), we omit the details as the proof follows exactly the same steps as in the
proof of (3.7) with the exception that (3.3) is employed with (a, b, c, z) =
(
q, q3, q6, q2m
)
rather
than (a, b, c, z) =
(
q3, q, q6, q2m
)
. ■
Lemma 3.4. There holds
∑
n≥0A
′(m,n)qn =
∑
n≥0(A(m,n)−A(m+ 2, n))qn.
Proof. We need the following contiguous relation which can be found in [17, equation (2.1)]:
2ϕ1
[
a, b
c
; q, z
]
− 2ϕ1
[
a, b
c
; q, qz
]
= z
(1− a)(1− b)
1− c
2ϕ1
[
qa, qb
qc
; q, z
]
. (3.9)
Then by (3.9) applied with q → q2 and (a, b, c, z) =
(
q−1, q, q4, qm
)
, we get∑
n≥0
(A(m,n)−A(m+ 2, n))qn
=
(
q2, q4; q2
)
∞(
q−1, q; q2
)
∞
(
2ϕ1
[
q−1, q
q4
; q2, qm
]
− 2ϕ1
[
q−1, q
q4
; q2, qm+2
])
=
(
q2, q4; q2
)
∞(
q−1, q; q2
)
∞
qm
(
1− q−1
)
(1− q)
1− q4
2ϕ1
[
q, q3
q6
; q2, qm
]
=
(
q2, q6; q2
)
∞(
q, q3; q2
)
∞
∑
n≥0
qm(n+1)
(
q, q3; q2
)
n(
q2, q6; q2
)
n
=
∑
n≥0
qm(n+1)
(
q2n+2; q2
)
∞
(
q2n+6; q2
)
∞(
q2n+1; q2
)
∞
(
q2n+3; q2
)
∞
.
This proves the lemma. ■
6 G.E. Andrews and M. El Bachraoui
Lemma 3.5. We have
∑
n≥0
B(2, n)qn =
−q3
(
1− q3 − q4 − q5
)(
1− q3
)2(
1− q5
) (3.10)
and for any positive integer m ≥ 2,
∑
n≥0
B(2m,n)qn =
1
(1− q)
(
1− q3
) ∑
n≥0
q3n+4
(
q2n+2; q2
)
m−1(
q2n+1; q2
)
m−2
. (3.11)
By (1.1) and (3.4), we obtain
∑
n≥0
B(2, n)q2n =
∑
n≥0
q2n
(
q2n+2, q2n+4; q2
)
∞(
q2n−3, q2n+3; q2
)
∞
=
(
q2, q4; q2
)
∞(
q−3, q3; q2
)
∞
∑
n≥0
(
q−3, q3; q2
)
n
q2n(
q2, q4; q2
)
n
=
(
q2, q4; q2
)
∞(
q−3, q3; q2
)
∞
2ϕ1
[
q−3, q3
q4
; q2, q2
]
=
(
q2, q4; q2
)
∞(
q−1, q; q2
)
∞
(
q, q5; q2
)
∞(
q2, q4; q2
)
∞
2ϕ1
[
q−2, q3
q5
; q2, q
]
=
q4
(1− q)
(
1− q3
)2
(
1 +
q
(
1− q−2
)(
1− q3
)(
1− q2
)(
1− q5
) )
=
q4
(1− q)
(
1− q3
)2 − q3
(1− q)
(
1− q3
)(
1− q5
) =
−q3
(
1− q3 − q4 − q5
)(
1− q3
)2(
1− q5
) .
This proves (3.10).
Now assume that m is a positive integer such that m ≥ 2. Then by using (1.1) and (3.3), we
have
∑
n≥0
B(2m,n)qn =
∑
n≥0
q2mn
(
q2n+2, q2n+4; q2
)
∞(
q2n−3, q2n+3; q2
)
∞
=
(
q2, q4; q2
)
∞(
q−3, q3; q2
)
∞
2ϕ1
[
q−3, q3
q4
; q2, q2m
]
=
(
q2, q4; q2
)
∞(
q−3, q3; q2
)
∞
(
q3, q2m−3; q2
)
∞(
q4, q2m; q2
)
∞
2ϕ1
[
q, q2m
q2m−3
; q2, q3
]
=
(
q2, q2m−3; q2
)
∞(
q−3, q2m; q2
)
∞
(
q, q2m; q2
)
∞(
q2, q2m−3; q2
)
∞
∑
n≥0
q3n
(
q2n+2, q2n+2m−3; q2
)
∞(
q2n+1, q2n+2m; q2
)
∞
=
q4
(1− q)
(
1− q3
) ∑
n≥0
q3n
(
q2n+2; q2
)
m−1(
q2n+1; q2
)
m−2
,
which yields (3.11).
Lemma 3.6. For any positive integer, we have
∑
n≥0
B′(2m,n)qn =
−1
1− q
∑
n≥0
q5n+2m+1
(
q2n+2; q2
)
m−1(
q2n+1; q2
)
m−1
.
Positive Weighted Partitions Generated by Double Series 7
Proof. By (1.1) and (3.3) applied to (a, b, c, z) =
(
q−1, q5, q6, q2m
)
, we find
∑
n≥0
B′(2m,n)qn = q2m
(
q2, q6; q2
)
∞(
q−1, q5; q2
)
∞
2ϕ1
[
q−1, q5
q6
; q2, q2m
]
= q2m
(
q2, q6; q2
)
∞(
q−1, q5; q2
)
∞
(
q5, q2m−1; q2
)
∞(
q6, q2m; q2
)
∞
2ϕ1
[
q, q2m
q2m−1
; q2, q5
]
= q2m
(
q2, q2m−1; q2
)
∞(
q−1, q2m; q2
)
∞
∑
n≥0
q5n
(
q, q2m; q2
)
n(
q2, q2m−1; q2
)
n
=
−1
1− q
∑
n≥0
q5n+2m+1
(
q2n+2, q2n+2m−1; q2
)
∞(
q2n+1, q2n+2m; q2
)
∞
=
−1
1− q
∑
n≥0
q5n+2m+1
(
q2n+2; q2
)
m−1(
q2n+1; q2
)
m−1
,
which is the desired identity. ■
Lemma 3.7. There holds
∑
n≥0B
′(m,n)qn =
∑
n≥0(B(m,n)−B(m+ 2, n))qn.
Proof. By (3.9) applied with q → q2 and (a, b, c, z) =
(
q−3, q3, q4, qm
)
, we get∑
n≥0
(B(m,n)−B(m+ 2, n))qn
=
(
q2, q4; q2
)
∞(
q−3, q3; q2
)
∞
(
2ϕ1
[
q−3, q3
q4
; q2, qm
]
− 2ϕ1
[
q−3, 3
q4
; q2, qm+2
])
=
(
q2, q4; q2
)
∞(
q−3, q3; q2
)
∞
qm
(
1− q−3
)(
1− q3
)
1− q4
2ϕ1
[
q−1, q5
q6
; q2, qm
]
=
(
q2, q6; q2
)
∞(
q−1, q5; q2
)
∞
∑
n≥0
qm(n+1)
(
q−1, q5; q2
)
n(
q2, q6; q2
)
n
=
∑
n≥0
qm(n+1)
(
q2n+2; q2
)
∞
(
q2n+6; q2
)
∞(
q2n−1; q2
)
∞
(
q2n+5; q2
)
∞
,
as desired. ■
4 Proof of Theorem 2.2 and Corollary 2.3
Proof of Theorem 2.2. By virtue of Lemma 3.4, we obtain∑
n≥0
A(2m+ 2, n)qn =
∑
n≥0
A(2m,n)qn −
∑
n≥0
A′(2m,n)qn
=
∑
n≥0
A(2m− 2, n)qn −
(∑
n≥0
A′(2m− 2, n)qn +
∑
n≥0
A′(2m,n)qn
)
· · ·
=
∑
n≥0
A(2, n)qn −
m∑
k=1
∑
n≥0
A′(2k, n)qn. (4.1)
Now combine (4.1) with Lemma 3.2 and (3.8) to deduce
−1
(1− q)2
∑
n≥0
qn+1
(
q2n+2; q2
)
m(
q2n+3; q2
)
m−1
=
−q
(
1 + q2
)
(1− q)2
(
1− q3
) − 1
1− q
∑
n≥0
m∑
k=1
q2k+3n
(
q2n+2; q2
)
k−1(
q2n+3; q2
)
k−1
,
8 G.E. Andrews and M. El Bachraoui
which by letting m→ ∞ yields
−q
(1− q)2
∑
n≥0
qn
(
q2n+2; q2
)
∞(
q2n+3; q2
)
∞
=
−q
(
1 + q2
)
(1− q)2
(
1− q3
) − 1
1− q
∑
n,k≥0
q2k+5n+2
(
q2n+2; q2
)
k(
q2n+1; q2
)
k
. (4.2)
In addition, by (3.5) and simplification we have∑
n≥0
qn
(
q2n+2; q2
)
∞(
q2n+3; q2
)
∞
=
(
q2; q2
)
∞(
q3; q2
)
∞
∑
n≥0
(
q3; q2
)
n
qn(
q2; q2
)
n
=
(
q2; q2
)
∞(
q3; q2
)
∞
(
q4; q2
)
∞(
q; q2
)
∞
=
1− q
1− q2
(
q2; q2
)2
∞(
q; q2
)2
∞
. (4.3)
Finally, combine (4.3) with (4.2) and rearrange to obtain
∑
n,k≥0
q2k+3n+2
(
q2n+2; q2
)
k(
q2n+3; q2
)
k
=
1
1− q2
(
q2; q2
)2
∞(
q; q2
)2
∞
− 1 + q2
(1− q)
(
1− q3
) ,
which gives the desired formula. ■
Proof of Corollary 2.3. An application of (3.6) with q → q4 and (a, b, z) =
(
q, q5, q
)
yields
after simplification
∞∑
n=−∞
(q; q4)nq
n
(q5; q4)n
= (1− q)
(
q2; q2
)2
∞(
q; q2
)2
∞
,
that is,
∑∞
n=−∞
qn
1−q4n+1 = (q2;q2)2∞
(q;q2)2∞
, or equivalently,
∞∑
n=0
(
qn
1− q4n+1
− q3n+2
1− q4n+3
)
=
(
q2; q2
)2
∞(
q; q2
)2
∞
. (4.4)
Thus, by using Theorem 2.2 and (4.4) we obtain
∑
n≥0
F1(n)q
n =
−
(
1 + q2
)
(1− q)
(
1− q3
) + 1
1− q2
(
q2; q2
)2
∞(
q; q2
)2
∞
=
−
(
1 + q2
)
(1− q)
(
1− q3
) + 1
1− q2
(
1
1− q
− q2
1− q3
)
+
∑
n≥1
(
qn
1− q4n+1
− q3n+2
1− q4n+3
)
1
1− q2
=
−q2
(1− q)
(
1− q3
) +∑
n≥1
(
qn
(
1− q2n
)(
1− q4n+1
)(
1− q2
) + q3n(
1− q4n+1
)(
1− q4n+3
))
⪰ −q2
(1− q)
(
1− q3
) +∑
n≥1
qn
(
1− q2n
)
1− q2
+
∑
n≥1
q3n
=
−q2
(1− q)
(
1− q3
) + q
(1− q)
(
1− q2
) − q3(
1− q2
)(
1− q3
) + q3
1− q3
=
q + q3
1− q3
⪰ 0.
This completes the proof. ■
Positive Weighted Partitions Generated by Double Series 9
5 Proof of Theorem 2.5 and Corollary 2.6
Proof of Theorem 2.5. Combining (4.1) with Lemma 3.2 and (3.7), we derive
1
(1− q)2
∑
n≥0
qn+1
(
q2n+2; q2
)
m(
q2n+3; q2
)
m−1
=
q
(
1 + q2
)
(1− q)2
(
1− q3
) + 1
1− q3
∑
n≥0
m∑
k=1
q2k+n
(
q2n+2; q2
)
k−1(
q2n+5; q2
)
k−1
,
which by letting m→ ∞ implies
1
(1− q)2
∑
n≥0
qn+1
(
q2n+2; q2
)
∞(
q2n+3; q2
)
∞
=
q
(
1 + q2
)
(1− q)2
(
1− q3
) + 1
1− q3
∑
n,k≥0
q2k+2+n
(
q2n+2; q2
)
k(
q2n+5; q2
)
k
,
or equivalently,
∑
n,k≥0
q2k+2+n
(
q2n+2; q2
)
k(
q2n+5; q2
)
k
=
1− q3
(1− q)2
∑
n≥0
qn+1
(
q2n+2; q2
)
∞(
q2n+3; q2
)
∞
−
q
(
1 + q2
)
(1− q)2
.
Now using (4.3), we derive
∑
n,k≥0
q2k+2+n
(
q2n+2; q2
)
k(
q2n+5; q2
)
k
=
q
(
1− q3
)
(1− q)2
1− q
1− q2
(
q2; q2
)2
∞(
q; ; q2
)2
∞
−
q
(
1 + q2
)
(1− q)2
,
which clearly is equivalent to the desired formula. ■
Proof of Corollary 2.6. From Theorem 2.5 and (4.4), we obtain
∑
n≥0
F2(n)q
n =
−q
(
1 + q2
)
(1− q)2
+
q
(
1− q3
)
(1− q)
(
1− q2
) (q2; q2)2∞(
q; q2
)2
∞
=
−q
(
1 + q2
)
(1− q)2
+
q
(
1− q3
)
(1− q)
(
1− q2
) ( 1− q2
(1− q)
(
1− q3
)
+
∑
n≥1
(
qn
1− q4n+1
− q3n+2
1− q4n+3
)
=
−q3
(1− q)2
+
∑
n≥1
(
qn
(
1− q2n
)(
1− q4n+1
)(
1− q2
) + q3n(
1− q4n+1
)(
1− q4n+3
))
⪰ −q3
(1− q)2
+
∑
n≥1
(
qn
(
1− q2n
)
1− q2
+ q3n
)
=
−q3
(1− q)2
+
q
(1− q)
(
1− q2
) − q3(
1− q2
)(
1− q3
) + q3
1− q3
=
−q3
(1− q)2
+
q
(1− q)
(
1− q2
) + q3 − q4
(1− q)
(
1− q3
)
=
2− 2q4 + q5
(1− q)
(
1− q3
) =
q
(
1− q3
)2
+ q5 − q7
(1− q)
(
1− q3
)
= q
(
1 + q + q2
)
+
q5(1 + q)
1− q3
⪰ 0.
This gives the desired result. ■
10 G.E. Andrews and M. El Bachraoui
6 Proof of Theorem 2.8 and Corollary 2.9
Proof of Theorem 2.8. By Lemma 3.7, we get∑
n≥0
B(2m+ 2, n)qn =
∑
n≥0
B(2m,n)qn −
∑
n≥0
B′(2m,n)qn
=
∑
n≥0
B(2m− 2, n)qn −
(∑
n≥0
B′(2m− 2, n)qn +
∑
n≥0
B′(2m,n)qn
)
· · ·
=
∑
n≥0
B(2, n)qn −
m∑
k=1
∑
n≥0
B′(2k, n)qn. (6.1)
Now use (6.1), Lemmas 3.5 and 3.6 to deduce
1
(1− q)
(
1− q3
) ∑
n≥0
q3n+4
(
q2n+2; q2
)
m(
q2n+1; q2
)
m−1
=
q3
(
−1 + q3 + q4 + q5
)(
1− q3
)2(
1− q5
) +
q
1− q
∑
n≥0
m∑
k=1
q2k+5n
(
q2n+2; q2
)
k−1(
q2n+1; q2
)
k−1
,
which by letting m→ ∞ and using (3.5) gives
∑
n,k≥0
q2k+3n+2
(
q2n+2; q2
)
k(
q2n+3; q2
)
k
=
q3
1− q3
∑
n≥0
q3n
(
q2n+2; q2
)
∞(
q2n+1; q2
)
∞
−
q2(1− q)
(
−1 + q3 + q4 + q5
)(
1− q3
)2(
1− q5
)
=
q3
1− q3
(
q2; q2
)
∞(
q; q2
)
∞
(
q4; q2
)
∞(
q3; q2
)
∞
−
q2(1− q)
(
−1 + q3 + q4 + q5
)(
1− q3
)2(
1− q5
)
=
q3(1− q)(
1− q2
)(
1− q3
) (q2; q2)2∞(
q; q2
)2
∞
−
q2(1− q)
(
−1 + q3 + q4 + q5
)(
1− q3
)2(
1− q5
)
=
q3
(1 + q)
(
1− q3
) (q2; q2)2∞(
q; q2
)2
∞
−
q2(1− q)
(
−1 + q3 + q4 + q5
)(
1− q3
)2(
1− q5
) ,
which is the desired identity. ■
Proof of Corollary 2.9. By virtue of Theorem 2.8 and (4.4), we have
∑
n≥0
G(n)qn =
q3
(1 + q)
(
1− q3
) (q2; q2)2∞(
q; q2
)2
∞
−
q2(1− q)
(
−1 + q3 + q4 + q5
)(
1− q3
)2(
1− q5
)
=
q3
(1 + q)
(
1− q3
) ( 1
1− q
− q2
1− q3
)
+
q3
(1 + q)
(
1− q3
) ∑
n≥1
(
qn
1− q4n+1
− q3n+2
1− q4n+3
)
−
q2(1− q)
(
−1 + q3 + q4 + q5
)(
1− q3
)2(
1− q5
)
=
q2(
1− q3
)(
1− q5
) +∑
n≥1
qn+3 − q3n+5 − q5n+6 + q7n+6
(1 + q)
(
1− q3
)(
1− q4n+1
)(
1− q4n+3
)
Positive Weighted Partitions Generated by Double Series 11
=
q2(
1− q3
)(
1− q5
) +∑
n≥1
qn+3
(
1− q2n+2
)
− q5n+6
(
1− q2n
)
(1 + q)
(
1− q3
)(
1− q4n+1
)(
1− q4n+3
)
⪰ q2(
1− q3
)(
1− q5
) +∑
n≥1
qn+3
(
1− q2n+2
)
(1 + q)
(
1− q3
) −
∑
n≥1
q5n+6
(
1− q2n
)
(1 + q)
(
1− q3
)
=
q2(
1− q3
)(
1− q5
) + q3
(1 + q)
(
1− q3
) ( 1
1− q
− q2
1− q3
)
+
q6
(1 + q)
(
1− q3
) ( −1
1− q5
+
1
1− q7
)
=
q2(
1− q3
)(
1− q5
) + q3(
1− q3
)2 − q11(1− q)(
1− q3
)(
1− q5
)(
1− q7
)
⪰ q2(
1− q3
)(
1− q5
) − q11(1− q)(
1− q3
)(
1− q5
)(
1− q7
)
=
q2 − q9 − q11 + q12(
1− q3
)(
1− q5
)(
1− q7
) =
q2
(
1− q3
)(
1− q7
)
+ q5 − q11(
1− q3
)(
1− q5
)(
1− q7
)
=
q2
1− q5
+
q5
(
1 + q3
)(
1− q5
)(
1− q7
) =
q2
1− q5
+
q5
(
1− q6
)(
1− q3
)(
1− q5
)(
1− q7
) ⪰ 0,
as desired. ■
Acknowledgements
First author partially supported by Simons Foundation Grant 633284. The authors are grateful
to two anonymous referees for their valuable comments and interesting suggestions which have
improved the presentation and quality of the paper.
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http://arxiv.org/abs/1205.4340
http://euler.univ-lyon1.fr/home/kratt/papers.html
https://doi.org/10.5281/zenodo.10623213
http://arxiv.org/abs/1901.01993
https://doi.org/10.1016/j.aim.2021.108028
http://arxiv.org/abs/1901.10886
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http://arxiv.org/abs/1003.1999
1 Introduction
2 Main results
3 Preliminary lemmas
4 Proof of Theorem 2.2 and Corollary 2.3
5 Proof of Theorem 2.5 and Corollary 2.6
6 Proof of Theorem 2.8 and Corollary 2.9
References
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| id | nasplib_isofts_kiev_ua-123456789-213520 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:50:11Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Andrews, George E. El Bachraoui, Mohamed 2026-02-18T11:23:50Z 2025 Positive Weighted Partitions Generated by Double Series. George E. Andrews and Mohamed El Bachraoui. SIGMA 21 (2025), 056, 12 pages 1815-0659 2020 Mathematics Subject Classification: 11P81; 05A17; 11D09 arXiv:2503.10890 https://nasplib.isofts.kiev.ua/handle/123456789/213520 https://doi.org/10.3842/SIGMA.2025.056 We investigate some weighted integer partitions whose generating functions are double-series. We will establish closed formulas for these -double series and deduce that their coefficients are non-negative. This leads to inequalities among integer partitions. First author partially supported by Simons Foundation Grant 633284. The authors are grateful to two anonymous referees for their valuable comments and interesting suggestions, which have improved the presentation and quality of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Positive Weighted Partitions Generated by Double Series Article published earlier |
| spellingShingle | Positive Weighted Partitions Generated by Double Series Andrews, George E. El Bachraoui, Mohamed |
| title | Positive Weighted Partitions Generated by Double Series |
| title_full | Positive Weighted Partitions Generated by Double Series |
| title_fullStr | Positive Weighted Partitions Generated by Double Series |
| title_full_unstemmed | Positive Weighted Partitions Generated by Double Series |
| title_short | Positive Weighted Partitions Generated by Double Series |
| title_sort | positive weighted partitions generated by double series |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213520 |
| work_keys_str_mv | AT andrewsgeorgee positiveweightedpartitionsgeneratedbydoubleseries AT elbachraouimohamed positiveweightedpartitionsgeneratedbydoubleseries |