Czechoslovak Mathematical Journal, first online, pp. 1-9


$q$-analogues of two supercongruences of Z.-W. Sun

Cheng-Yang Gu, Victor J. W. Guo

Received November 26, 2018.   Published online January 22, 2020.

Abstract:  We give several different $q$-analogues of the following two congruences of Z.-W. Sun: $\sum_{k=0}^{(p^r-1)/2}\frac1{8^k}{2k\choose k} \equiv\Bigl(\frac2{p^r}\Bigr)\pmod{p^2}$ and $\sum_{k=0}^{(p^r-1)/2}\frac1{16^k}{2k\choose k}\equiv\Bigl(\frac3{p^r}\Bigr)\pmod{p^2}$, where $p$ is an odd prime, $r$ is a positive integer, and $(\frac mn)$ is the Jacobi symbol. The proofs of them require the use of some curious $q$-series identities, two of which are related to Franklin's involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.
Keywords:  congruences; $q$-binomial coefficient; cyclotomic polynomial; Franklin's involution
Classification MSC:  11B65, 05A10, 05A30, 11A07
DOI:  10.21136/CMJ.2020.0516-18

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References:
[1] G. E. Andrews: The Theory of Partitions. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511608650 | MR 1634067 | Zbl 0996.11002
[2] A. Berkovich, F. G. Garvan: Some observations on Dyson's new symmetries of partitions. J. Comb. Theory, Ser. A 100 (2002), 61-93. DOI 10.1006/jcta.2002.3281 | MR 1932070 | Zbl 1016.05003
[3] J. Cigler: A new class of $q$-Fibonacci polynomials. Electron. J. Comb. 10 (2003), Research paper R19, 15 pages. MR 1975769 | Zbl 1027.05006
[4] S. B. Ekhad, D. Zeilberger: The number of solutions of $X^2=0$ in triangular matrices over $GF(q)$. Electron. J. Comb. 3 (1996), Research paper R2, 2 pages. MR 1364064 | Zbl 0851.15010
[5] V. J. W. Guo: Common $q$-analogues of some different supercongruences. Result. Math. 74 (2019), Article No. 131, 15 pages. DOI 10.1007/s00025-019-1056-1 | MR 3963751 | Zbl 1414.33016
[6] V. J. W. Guo, J.-C. Liu: $q$-analogues of two Ramanujan-type formulas for $1/\pi$. J. Difference Equ. Appl. 24 (2018), 1368-1373. DOI 10.1080/10236198.2018.1485669 | MR 3851167 | Zbl 06949015
[7] V. J. W. Guo, S.-D. Wang: Factors of sums and alternating sums of products of $q$-binomial coefficients and powers of $q$-integers. Taiwanese J. Math. 23 (2019), 11-27. DOI 10.11650/tjm/180601 | MR 3909988 | Zbl 1405.05017
[8] V. J. W. Guo, J. Zeng: New congruences for sums involving Apéry numbers or central Delannoy numbers. Int. J. Number Theory 8 (2012), 2003-2016. DOI 10.1142/S1793042112501138 | MR 2978852 | Zbl 1268.11028
[9] V. J. W. Guo, W. Zudilin: A $q$-microscope for supercongruences. Adv. Math. 346 (2019), 329-358. DOI 10.1016/j.aim.2019.02.008 | MR 3910798 | Zbl 07035902
[10] M. E. H. Ismail, D. Kim, D. Stanton: Lattice paths and positive trigonometric sums. Constructive Approximation 15 (1999), 69-81. DOI 10.1007/s003659900097 | MR 1660081 | Zbl 0924.42004
[11] J.-C. Liu: Some finite generalizations of Euler's pentagonal number theorem. Czech. Math. J. 67 (2017), 525-531. DOI 10.21136/CMJ.2017.0063-16 | MR 3661057 | Zbl 06738535
[12] J.-C. Liu: Some finite generalizations of Gauss's square exponent identity. Rocky Mt. J. Math. 47 (2017), 2723-2730. DOI 10.1216/RMJ-2017-47-8-2723 | MR 3760315 | Zbl 06840997
[13] L. J. Slater: A new proof of Rogers's transformations of infinite series. Proc. Lond. Math. Soc., II. Ser. 53 (1951), 460-475. DOI 10.1112/plms/s2-53.6.460 | MR 0043235 | Zbl 0044.06102
[14] Z.-W. Sun: Fibonacci numbers modulo cubes of primes. Taiwanese J. Math. 17 (2013), 1523-1543. DOI 10.11650/tjm.17.2013.2488 | MR 3106028 | Zbl 1316.11013

Affiliations:   Cheng-Yang Gu, Victor J. W. Guo (corresponding author), School of Mathematics and Statistics, Huaiyin Normal University, Huai'an 223300, Jiangsu, P. R. China, e-mail: 525290408@qq.com, jwguo@hytc.edu.cn


 
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