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Proof of a q-supercongruence conjectured by Guo and Schlosser
In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer n > 1 and M = ( n + 1 ) / 2 or n - 1 , ∑ k = 0 M [ 4 k - 1 ] q 2 [ 4 k - 1 ] 2 ( q - 2 ; q 4 ) k 4 ( q 4 ; q 4 ) k 4 q 4 k ≡ ( 2 q + 2 q - 1 - 1 ) [ n ] q 2 4 ( mod [ n ] q 2 4 Φ n ( q 2 ) ) , where [ n ]...
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| Published in: | Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2020-10, Vol.114 (4), Article 190 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer
n
>
1
and
M
=
(
n
+
1
)
/
2
or
n
-
1
,
∑
k
=
0
M
[
4
k
-
1
]
q
2
[
4
k
-
1
]
2
(
q
-
2
;
q
4
)
k
4
(
q
4
;
q
4
)
k
4
q
4
k
≡
(
2
q
+
2
q
-
1
-
1
)
[
n
]
q
2
4
(
mod
[
n
]
q
2
4
Φ
n
(
q
2
)
)
,
where
[
n
]
=
[
n
]
q
=
(
1
-
q
n
)
/
(
1
-
q
)
,
(
a
;
q
)
0
=
1
,
(
a
;
q
)
k
=
(
1
-
a
)
(
1
-
a
q
)
⋯
(
1
-
a
q
k
-
1
)
for
k
≥
1
and
Φ
n
(
q
)
denotes the
n
-th cyclotomic polynomial. |
|---|---|
| ISSN: | 1578-7303 1579-1505 |
| DOI: | 10.1007/s13398-020-00923-2 |