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Some congruences involving fourth powers of central q-binomial coefficients

Part of: Sequences and sets Combinatorics

Published online by Cambridge University Press:  30 January 2019

Victor J. W. Guo  and
Su-Dan Wang
Victor J. W. Guo
Affiliation:
School of Mathematical Sciences, Huaiyin Normal University, Huai'an, Jiangsu223300, People's Republic of China (jwguo@hytc.edu.cn)
Su-Dan Wang
Affiliation:
Department of Mathematics, East China Normal University, Shanghai200062, People's Republic of China (sudan199219@126.com)
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Abstract

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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.

Keywords

q-binomial coefficientsq-WZ methodcyclotomic polynomialsq-analogue of Wolstenholme's binomial congruenceq-analogue of Morley's congruence

MSC classification

Primary: 11B65: Binomial coefficients; factorials; $q$-identities
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions 05A30: $q$-calculus and related topics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1127 - 1138
Copyright
Copyright © 2019 The Royal Society of Edinburgh

References

1Andrews, G. E.. The theory of partitions (Cambridge: Cambridge University Press, 1998).Google Scholar
2Babbage, C.. Demonstration of a theorem relating to prime numbers. Edinburgh Philos. J. 1 (1819), 46–49.Google Scholar
3Chowla, S., Dwork, B. and Evans, R.. On the mod p 2 determination of $(p-1)/2\choose (p-1)/4$. J. Number Theory (1986), 188–196.CrossRefGoogle Scholar
4Gasper, G. and Rahman, M.. Basic hypergeometric series. 2nd Edn, Encyclopedia of Mathematics and its applications, vol. 96 (Cambridge: Cambridge University Press, 2004).10.1017/CBO9780511526251CrossRefGoogle Scholar
5Guo, V. J. W.. Some congruences related to the q-Fermat quotients. Int. J. Number Theory 11 (2015), 1049–1060.10.1142/S1793042115500554CrossRefGoogle Scholar
6Guo, V. J. W.. Some generalizations of a supercongruence of van Hamme. Integral Transforms Spec. Funct. 28 (2017), 888–899.CrossRefGoogle Scholar
7Guo, V. J. W.. q-Analogues of the (E.2) and (F.2) supercongruences of Van Hamme. Ramanujan J. (2018), https://doi.org/10.1007/s11139-018-0021-zGoogle Scholar
8Guo, V. J. W.. A q-analogue of a Ramanujan-type supercongruence involving central binomial coefficients. J. Math. Anal. Appl. 458 (2018), 590–600.CrossRefGoogle Scholar
9Koepf, W.. Hypergeometric summation, an Algorithmic approach to summation and special function identities, 2nd Edn (London: Springer, 2014).CrossRefGoogle Scholar
10Liu, J., Pan, H. and Zhang, Y.. A generalization of Morley's congruence. Adv. Differ. Equ. 2015 (2015), 254.10.1186/s13662-015-0568-6CrossRefGoogle Scholar
11Long, L.. Hypergeometric evaluation identities and supercongruences. Pacific J. Math. 249 (2011), 405–418.CrossRefGoogle Scholar
12Městrović, R.. Wolstenholme's theorem: its generalizations and extensions in the last hundred and fifty years (1862–2012), preprint, 2011, arXiv:1111.3057.Google Scholar
13Morley, F.. Note on the congruence 24n≡(−1)n (2n)!/(n!)2, where 2n + 1 is a prime. Ann. of Math. 9 (1895), 168–170.10.2307/1967516CrossRefGoogle Scholar
14Mortenson, E.. A p-adic supercongruence conjecture of van Hamme. Proc. Amer. Math. Soc. 136 (2008), 4321–4328.10.1090/S0002-9939-08-09389-1CrossRefGoogle Scholar
15Pan, H.. A q-analogue of Lehmer's congruence. Acta Arith. 128 (2007), 303–318.10.4064/aa128-4-1CrossRefGoogle Scholar
16Pan, H.. An elementary approach to $(p-1)/2\choose (p-1)/4$ modulo p 2. Taiwanese J Math. 16 (2012), 2197–2202.10.11650/twjm/1500406847CrossRefGoogle Scholar
17Petkovšek, M., Wilf, H. S. and Zeilberger, D.. A = B (Wellesley, MA: A K Peters, Ltd., 1996).10.1201/9781439864500CrossRefGoogle Scholar
18Shi, L.-L. and Pan, H.. A q-analogue of Wolstenholme's harmonic series congruence. Amer. Math. Monthly 114 (2005), 529–531.10.1080/00029890.2007.11920441CrossRefGoogle Scholar
19Straub, A.. A q-analog of Ljunggren's binomial congruence, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011). Discrete Math. Theor. Comput. Sci. Proc., AO, Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2011) 897–902.Google Scholar
20Sun, Z.-W.. Products and sums divisible by central binomial coefficients. Electron. J. Combin. 20 (2013), #P9.10.37236/3022CrossRefGoogle Scholar
21Swisher, H.. On the supercongruence conjectures of van Hamme. Res. Math. Sci. (2015) 2–18.Google Scholar
22Tauraso, R.. Some q-analogs of congruences for central binomial sums. Colloq. Math. 133 (2013), 133–143.10.4064/cm133-1-9CrossRefGoogle Scholar
23van Hamme, L.. Some conjectures concerning partial sums of generalized hypergeometric series, p-adic functional analysis (Nijmegen, 1996), Lecture Notes in Pure and Appl. Math.,vol. 192 (New York: Dekker, 1997), 223–236.Google Scholar
24Wang, S.-D.. Generalizations of a supercongruence involving $\left( {\matrix{ {2k} \cr k \cr}} \right)^4$. J. Difference Equ. Appl. 24 (2018), 1375–1383.10.1080/10236198.2018.1485667CrossRefGoogle Scholar
25Wolstenholme, J.. On certain properties of prime numbers. Quart. J. Pure Appl. Math. 5 (1862), 35–39.Google Scholar
26Zudilin, W.. Ramanujan-type supercongruences. J. Number Theory 129 (2009), 1848–1857.CrossRefGoogle Scholar