Some More Identities of Kanade–Russell Type Derived Using Rosengren’s Method
In the present paper, we consider some variations and generalizations of the multi-sum to single-sum transformation recently used by Rosengren in his proof of the Kanade–Russell identities. These general transformations are then used to prove a number of identities equating multi-sums and infinite p...
Gespeichert in:
| Veröffentlicht in: | Annals of combinatorics 2023-06, Vol.27 (2), p.329-352 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Zusammenfassung: | In the present paper, we consider some variations and generalizations of the multi-sum to single-sum transformation recently used by Rosengren in his proof of the Kanade–Russell identities. These general transformations are then used to prove a number of identities equating multi-sums and infinite products or multi-sums and infinite product
×
a false theta series. Examples include the following:
∑
j
,
k
,
p
,
r
=
0
∞
(
-
1
)
j
+
k
q
(
2
j
+
k
-
p
+
r
)
2
/
2
+
k
(
k
+
4
)
/
2
+
3
j
-
p
/
2
+
3
r
/
2
(
-
q
;
q
)
r
q
2
;
q
2
j
(
q
;
q
)
k
(
q
;
q
)
p
(
q
;
q
)
r
=
2
-
q
;
q
2
∞
-
q
2
,
-
q
14
,
q
16
;
q
16
∞
(
q
;
q
)
∞
.
Let
Q
(
i
,
j
,
k
,
l
,
p
)
:
=
1
2
(
i
+
6
j
+
4
k
+
2
l
-
p
)
(
i
+
6
j
+
4
k
+
2
l
-
p
-
1
)
+
2
k
(
k
-
1
)
+
l
(
l
-
1
)
+
3
i
+
15
j
+
14
k
+
5
l
-
2
p
.
Then
∑
i
,
j
,
k
,
l
,
p
=
0
∞
(
-
1
)
l
+
k
q
Q
(
i
,
j
,
k
,
l
,
p
)
q
;
q
i
q
6
;
q
6
j
q
4
;
q
4
k
q
2
;
q
2
l
(
q
;
q
)
p
=
2
(
-
q
;
q
)
∞
2
q
q
3
;
q
6
∞
q
4
;
q
4
∞
1
+
∑
r
=
1
∞
q
9
r
2
+
6
r
-
q
9
r
2
-
6
r
.
∑
j
,
k
,
p
=
0
∞
(
-
1
)
k
q
(
3
j
+
2
k
-
p
)
(
3
j
+
2
k
-
p
-
1
)
/
2
+
k
(
k
-
1
)
-
p
+
6
j
+
6
k
(
q
3
;
q
3
)
j
(
q
2
;
q
2
)
k
(
q
;
q
)
p
=
(
-
1
;
q
)
∞
(
q
18
;
q
18
)
∞
(
q
3
;
q
3
)
∞
(
q
9
;
q
18
)
∞
. |
|---|---|
| ISSN: | 0218-0006 0219-3094 |
| DOI: | 10.1007/s00026-022-00586-3 |