# American Institute of Mathematical Sciences

June  2020, 28(2): 1031-1036. doi: 10.3934/era.2020055

## A family of $q$-congruences modulo the square of a cyclotomic polynomial

 School of Mathematics and Statistics, Huaiyin Normal University, Huai'an 223300, Jiangsu, China

Received  January 2020 Revised  April 2020 Published  June 2020

Fund Project: The author was partially supported by the National Natural Science Foundation of China (grant 11771175)

Using Watson's terminating $_8\phi_7$ transformation formula, we prove a family of $q$-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636-646]. As an application, we deduce two supercongruences modulo $p^4$ ($p$ is an odd prime) and their $q$-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.

Citation: Victor J. W. Guo. A family of $q$-congruences modulo the square of a cyclotomic polynomial. Electronic Research Archive, 2020, 28 (2) : 1031-1036. doi: 10.3934/era.2020055
##### References:
 [1] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511526251.  Google Scholar [2] C.-Y. Gu and V. J. W. Guo, $q$-Analogues of two supercongruences of Z.-W. Sun, Czechoslovak Math. J., in press. doi: 10.21136/CMJ.2020.0516-18.  Google Scholar [3] V. J. W. Guo, Common $q$-analogues of some different supercongruences, Results Math., 74 (2019), 15pp. doi: 10.1007/s00025-019-1056-1.  Google Scholar [4] V. J. W. Guo, Proof of a generalization of the (B.2) supercongruence of Van Hamme through a $q$-microscope, Adv. in Appl. Math., 116 (2020), 19pp. doi: 10.1016/j.aam.2020.102016.  Google Scholar [5] V. J. W. Guo, $q$-Analogues of Dwork-type supercongruences, J. Math. Anal. Appl., 487 (2020), 9pp. doi: 10.1016/j.jmaa.2020.124022.  Google Scholar [6] V. J. W. Guo and 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.  Google Scholar [7] V. J. W. Guo and M. J. Schlosser, Some new $q$-congruences for truncated basic hypergeometric series: Even powers, Results Math., 75 (2020), 15pp. doi: 10.1007/s00025-019-1126-4.  Google Scholar [8] V. J. W. Guo and J. Zeng, Some $q$-supercongruences for truncated basic hypergeometric series, Acta Arith., 171 (2015), 309-326.  doi: 10.4064/aa171-4-2.  Google Scholar [9] V. J. W. Guo and W. Zudilin, A $q$-microscope for supercongruences, Adv. Math., 346 (2019), 329-358.  doi: 10.1016/j.aim.2019.02.008.  Google Scholar [10] V. J. W. Guo and W. Zudilin, On a $q$-deformation of modular forms, J. Math. Anal. Appl., 475 (2019), 1636-1646.  doi: 10.1016/j.jmaa.2019.03.035.  Google Scholar [11] V. J. W. Guo and W. Zudilin, A common $q$-analogue of two supercongruences, Results Math., 75 (2020), 11pp. doi: 10.1007/s00025-020-1168-7.  Google Scholar [12] J.-C. Liu, Some supercongruences on truncated $_3F_2$ hypergeometric series, J. Difference Equ. Appl., 24 (2018), 438-451.  doi: 10.1080/10236198.2017.1418863.  Google Scholar [13] J.-C. Liu, On Van Hamme's (A.2) and (H.2) supercongruences, J. Math. Anal. Appl., 471 (2019), 613-622.  doi: 10.1016/j.jmaa.2018.10.095.  Google Scholar [14] L. Long and R. Ramakrishna, Some supercongruences occurring in truncated hypergeometric series, Adv. Math., 290 (2016), 773-808.  doi: 10.1016/j.aim.2015.11.043.  Google Scholar [15] H.-X. Ni and H. Pan, Some symmetric $q$-congruences modulo the square of a cyclotomic polynomial, J. Math. Anal. Appl., 481 (2020), 12pp. doi: 10.1016/j.jmaa.2019.07.062.  Google Scholar [16] Z.-H. Sun, Generalized Legendre polynomials and related supercongruences, J. Number Theory, 143 (2014), 293-319.  doi: 10.1016/j.jnt.2014.04.012.  Google Scholar [17] Z.-W. Sun, On sums of Apéry polynomials and related congruences, J. Number Theory, 132 (2012), 2673-2699.  doi: 10.1016/j.jnt.2012.05.014.  Google Scholar [18] H. Swisher, On the supercongruence conjectures of van Hamme, Res. Math. Sci., 2 (2015), 21pp. doi: 10.1186/s40687-015-0037-6.  Google Scholar [19] L. Van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, in $p$-Adic Functional Analysis, Lecture Notes in Pure and Appl. Math., 192, Dekker, New York, 1997,223–236.  Google Scholar [20] C. Wang and H. Pan, On a conjectural congruence of Guo, preprint, arXiv: 2001.08347. Google Scholar

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##### References:
 [1] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511526251.  Google Scholar [2] C.-Y. Gu and V. J. W. Guo, $q$-Analogues of two supercongruences of Z.-W. Sun, Czechoslovak Math. J., in press. doi: 10.21136/CMJ.2020.0516-18.  Google Scholar [3] V. J. W. Guo, Common $q$-analogues of some different supercongruences, Results Math., 74 (2019), 15pp. doi: 10.1007/s00025-019-1056-1.  Google Scholar [4] V. J. W. Guo, Proof of a generalization of the (B.2) supercongruence of Van Hamme through a $q$-microscope, Adv. in Appl. Math., 116 (2020), 19pp. doi: 10.1016/j.aam.2020.102016.  Google Scholar [5] V. J. W. Guo, $q$-Analogues of Dwork-type supercongruences, J. Math. Anal. Appl., 487 (2020), 9pp. doi: 10.1016/j.jmaa.2020.124022.  Google Scholar [6] V. J. W. Guo and 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.  Google Scholar [7] V. J. W. Guo and M. J. Schlosser, Some new $q$-congruences for truncated basic hypergeometric series: Even powers, Results Math., 75 (2020), 15pp. doi: 10.1007/s00025-019-1126-4.  Google Scholar [8] V. J. W. Guo and J. Zeng, Some $q$-supercongruences for truncated basic hypergeometric series, Acta Arith., 171 (2015), 309-326.  doi: 10.4064/aa171-4-2.  Google Scholar [9] V. J. W. Guo and W. Zudilin, A $q$-microscope for supercongruences, Adv. Math., 346 (2019), 329-358.  doi: 10.1016/j.aim.2019.02.008.  Google Scholar [10] V. J. W. Guo and W. Zudilin, On a $q$-deformation of modular forms, J. Math. Anal. Appl., 475 (2019), 1636-1646.  doi: 10.1016/j.jmaa.2019.03.035.  Google Scholar [11] V. J. W. Guo and W. Zudilin, A common $q$-analogue of two supercongruences, Results Math., 75 (2020), 11pp. doi: 10.1007/s00025-020-1168-7.  Google Scholar [12] J.-C. Liu, Some supercongruences on truncated $_3F_2$ hypergeometric series, J. Difference Equ. Appl., 24 (2018), 438-451.  doi: 10.1080/10236198.2017.1418863.  Google Scholar [13] J.-C. Liu, On Van Hamme's (A.2) and (H.2) supercongruences, J. Math. Anal. Appl., 471 (2019), 613-622.  doi: 10.1016/j.jmaa.2018.10.095.  Google Scholar [14] L. Long and R. Ramakrishna, Some supercongruences occurring in truncated hypergeometric series, Adv. Math., 290 (2016), 773-808.  doi: 10.1016/j.aim.2015.11.043.  Google Scholar [15] H.-X. Ni and H. Pan, Some symmetric $q$-congruences modulo the square of a cyclotomic polynomial, J. Math. Anal. Appl., 481 (2020), 12pp. doi: 10.1016/j.jmaa.2019.07.062.  Google Scholar [16] Z.-H. Sun, Generalized Legendre polynomials and related supercongruences, J. Number Theory, 143 (2014), 293-319.  doi: 10.1016/j.jnt.2014.04.012.  Google Scholar [17] Z.-W. Sun, On sums of Apéry polynomials and related congruences, J. Number Theory, 132 (2012), 2673-2699.  doi: 10.1016/j.jnt.2012.05.014.  Google Scholar [18] H. Swisher, On the supercongruence conjectures of van Hamme, Res. Math. Sci., 2 (2015), 21pp. doi: 10.1186/s40687-015-0037-6.  Google Scholar [19] L. Van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, in $p$-Adic Functional Analysis, Lecture Notes in Pure and Appl. Math., 192, Dekker, New York, 1997,223–236.  Google Scholar [20] C. Wang and H. Pan, On a conjectural congruence of Guo, preprint, arXiv: 2001.08347. Google Scholar
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