
doi: 10.1017/prm.2018.96
AbstractWe prove some congruences on sums involving fourth powers of central q-binomial coefficients. As a conclusion, we confirm the following supercongruence observed by Long [Pacific J. Math. 249 (2011), 405–418]: $$\sum\limits_{k = 0}^{((p^r-1)/(2))} {\displaystyle{{4k + 1} \over {{256}^k}}} \left( \matrix{2k \cr k} \right)^4\equiv p^r\quad \left( {\bmod p^{r + 3}} \right),$$where p⩾5 is a prime and r is a positive integer. Our method is similar to but a little different from the WZ method used by Zudilin to prove Ramanujan-type supercongruences.
\(q\)-analogue of Wolstenholme's binomial congruence, Binomial coefficients; factorials; \(q\)-identities, \(q\)-calculus and related topics, \(q\)-analogue of Morley's congruence, \(q\)-WZ method, Factorials, binomial coefficients, combinatorial functions, \(q\)-binomial coefficients, cyclotomic polynomials
\(q\)-analogue of Wolstenholme's binomial congruence, Binomial coefficients; factorials; \(q\)-identities, \(q\)-calculus and related topics, \(q\)-analogue of Morley's congruence, \(q\)-WZ method, Factorials, binomial coefficients, combinatorial functions, \(q\)-binomial coefficients, cyclotomic polynomials
| selected citations These citations are derived from selected sources. This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | 41 | |
| popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network. | Top 10% | |
| influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | Top 10% | |
| impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network. | Top 10% |
