Some q-supercongruences from Watson's 8φ7 Transformation Formula
Some q-supercongruences from Watson's 8φ7 Transformation Formula
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DOI：
10.1007/s00025-020-01195-3
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发表时间：
2020-04-24
影响因子：
2.2
通讯作者：
Yue, Mingbing
中科院分区：
文献类型：
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作者：
Wang, Xiaoxia;Yue, Mingbing
最近，Jana和Kalita证明了下列超同余：如果r是偶数，则Nk=0(-1)k(2dk+1)(1d)3k！3=Pr(mod Pr+2)；(-1)(p-d+1)Rd(d-1)Pr(mod Pr+2)，如果r是奇数；如果r是偶数，Nk=0(-1)k(2dk+1)3(1d)3k！3=-3Pr(mod Pr+2)，如果r是奇数，则证明了下列超同余：Nk=0(-1)k(2dk+1)(1d)3k！3=-3Pr(mod Pr+2)，如果r是偶数；(-1)(p+1)dr3(d-1)Pr(mod Pr+2)，如果r是奇数，其中N=Pr-1d，如果r是偶数；(d-1)Pr-1d，如果r是奇数。从Watson的8f7变换公式出发，给出了上述超同余的q-类比，推广了Van Hamme以前的一些猜想结果。我们的证明使用的是郭德纲和祖地林提出的“创造性显微镜”方法。
Recently, Jana and Kalita proved the following supercongruences involving rising factorials ( 1 d)3k: N k=0(-1)k(2dk + 1) ( 1 d)3 k k!3 = pr (mod pr+2), if r is even; (-1) (p-d+1)r d (d - 1)pr (mod pr+2), if r is odd; N k=0(-1)k(2dk + 1)3 ( 1 d)3 k k!3 = -3pr (mod pr+2), if r is even; (-1) (p+1)r d 3(d - 1)pr (mod pr+2), if r is odd, where N = pr- 1 d, if r is even; (d-1)pr-1 d, if r is odd. From Watson's 8f7 transformation formula, we give q-analogues of the above supercongruences, generalizing some previous conjectural results of Van Hamme. Our proof uses the `creative microscoping' method which was introduced by Guo and Zudilin.