Some congruences involving fourth powers of central q-binomial coefficients

By Victor J. W. Guo, Su-Dan Wang. Proceedings of the Royal Society of Edinburgh. Section A Mathematics, 2019

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Abstract We 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/53df3c6c81fff096616efb2b350f8ec337b8d700

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Citations in the corpus

Some congruences related to a congruence of Van Hamme (2019, methodology): “For more supercongruences and q-supercongruences, we refer the reader to [4, 6–20, 26, 29, 30].”
Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series (2018, background): “k=0 [4k + 1] (q; q(2))(4)k (q2; q(2))(4)k ≡ q(1−n)/2[n] + (n 2 − 1)(1− q)(2) 24 q(1−n)/2[n]3 (mod [n]Φn(q)(3)), due to the first author and Wang [27], is the unique q-congruence modulo [n]Φn(q) 3 in the literature that is completely proved.”
A family of q-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial (2019, background): “During the past few years, many congruences and supercongruences were generalized to the q-setting by various authors (see, for instance, [5–18, 25, 26, 30, 32]).”