Some new q -congruences on double sums
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Some new q-congruences on double sums Xiaoxia Wang1 · Menglin Yu1 Received: 28 April 2020 / Accepted: 8 October 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract Swisher confirmed an interesting congruence: for any odd prime p, ( p−1)/2 k=0
k ( 21 )3k 1 1 (−1) (6k + 1) 3 k − ≡0 k! 8 (2 j − 1)2 16 j 2 k
(mod p),
j=1
which was conjectured by Long. Recently, its q-analogue was proved by Gu and Guo. Inspired by their work, we obtain a new similar q-congruence modulo n (q) and two qsupercongruences modulo n (q)2 on double basic hypergeometric sums, where n (q) is the n-th cyclotomic polynomial. Keywords q-Congruence · Cyclotomic polynomial · Basic hypergeometric series · Supercongruence Mathematics Subject Classification Primary 33D15 · Secondary 11A07 · 11B65
1 Introduction It was in 1914 that Ramanujan [16] discovered a family of formulas for 1/π. These formulas lead to Ramanujan-type supercongruences or the p-adic analogues (see Van Hamme [19]), such as, for any odd prime p,
This work is supported by National Natural Science Foundations of China (11661032).
B
Menglin Yu [email protected] Xiaoxia Wang [email protected]
1
Department of Applied Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China 0123456789().: V,-vol
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Page 2 of 10 ( p−1)/2 k=0 ( p−1)/2 k=0 ( p−1)/3
X. Wang , M. Yu
(−1)k (6k + 1)
(mod p 3 );
(1.1)
( 21 )3k −1 (mod p 3 ); (−1) (4k + 1) 3 ≡ p k! p k
(−1)k (6k + 1)
( 13 )3k ≡p k!3
(−1)k (8k + 1)
( 41 )3k −2 ≡ p k!3 p
k=0 ( p−1)/4
( 21 )3k −2 ≡ p k!3 8k p
k=0
(1.2)
(mod p 3 ) if p ≡ 1 (mod 3); (mod p 3 ) if p ≡ 1 (mod 4).
(1.3)
(1.4)
where p· is the Legendre symbol modulo p and (a)n = a(a + 1) · · · (a + n − 1) denotes the Pochhammer symbol. It should be pointed out that all of these supercongruences have been proved with different methods up to now. In [17], Swisher gave proofs of some Van Hamme’s supercongruences, including (1.2), (1.3) and (1.4). Mortenson [14] proved the supercongruence (1.2) via a 6 F5 transformation. Later Zudilin [22] and Long [13] reproved (1.2) respectively. All in all, q-analogues of congruences and supercongruences have been studied by a lot of authors (see [2,5–8,11,12,15,20,21]). It is worth mentioning that Guo and Zudilin [10] introduced a method, called ‘creative microscoping’, which is powerful to prove q-supercongruences. Swisher [17, Corollary 1.4] confirmed an interesting congruence on double sums from (1.1) as follows: for any odd prime p, ( p−1)/2
(−1)k (6k + 1)
k=0
k ( 21 )3k 1 1 − ≡0 k!3 8k (2 j − 1)2 16 j 2
(mod p),
(1.5)
j=1
which was conjectured by Long [13]. Later Gu and Guo [3] gave a q-analogue of (1.5) as follows: for odd n, (n−1)/2
(−1)k [6k + 1]
k=0
k (q; q 2 )3k q 2 j−1 q4 j − ≡0 [2 j − 1]2 [4 j]2 (q 4 ; q 4 )3k
(mod n (q)),
(1.6)
j=1
which was originally conjectured by Guo [4, Conjecture 4.4]. Here and in what follows, we use some standard q-notations. The q-shifted factorial is given by (a; q)∞ =
(1 − aq j ),
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