# Article

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Keywords:
basic hypergeometric series; Watson's transformation; $q$-congruence; supercongruence; creative microscoping
Summary:
Let $\Phi _n(q)$ denote the $n$th cyclotomic polynomial in $q$. Recently, Guo, Schlosser and Zudilin proved that for any integer $n>1$ with $n\equiv 1\pmod {4}$, $$\sum _{k=0}^{n-1}\frac {(q^{-1};q^2)_k^2(q^{-2};q^4)_k}{(q^2;q^2)_k^2 (q^4;q^4)_k}q^{6k} \equiv 0\pmod {\Phi _n(q)^2},$$ where $(a;q)_m=(1-a)(1-aq)\cdots (1-aq^{m-1})$. In this note, we give a generalization of the above $q$-congruence to the modulus $\Phi _n(q)^3$ case. Meanwhile, we give a corresponding $q$-congruence modulo $\Phi _n(q)^2$ for $n\equiv 3\pmod {4}$. Our proof is based on the `creative microscoping' method, recently developed by Guo and Zudilin, and a $_4\varphi _3$ summation formula.
References:
[1] Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications 96. Cambridge University Press, Cambridge (2004). DOI 10.1017/CBO9780511526251 | MR 2128719 | Zbl 1129.33005
[2] Guo, V. J. W., Schlosser, M. J.: Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial. J. Difference Equ. Appl. 25 (2019), 921-929. DOI 10.1080/10236198.2019.1622690 | MR 3996958 | Zbl 1426.33048
[3] Guo, V. J. W., Schlosser, M. J.: Some $q$-supercongruences from transformation formulas for basic hypergeometric series. Constr. Approx. 53 (2021), 155-200. DOI 10.1007/s00365-020-09524-z | MR 4205256 | Zbl 07326825
[4] Guo, V. J. W., Schlosser, M. J., Zudili, W.: New quadratic identities for basic hypergeometric series and q-congruences. Preprint. Available at \let \relax \brokenlink{ http://math.ecnu.edu.cn/ jwguo/{maths/quad.pdf}}.
[5] Guo, V. J. W., Zudilin, W.: A $q$-microscope for supercongruences. Adv. Math. 346 (2019), 329-358. DOI 10.1016/j.aim.2019.02.008 | MR 3910798 | Zbl 07035902
[6] Guo, V. J. W., Zudilin, W.: On a $q$-deformation of modular forms. J. Math. Anal. Appl. 475 (2019), 1636-1646. DOI 10.1016/j.jmaa.2019.03.035 | MR 3944391 | Zbl 1445.11014
[7] Guo, V. J. W., Zudilin, W.: A common $q$-analogue of two supercongruences. Result. Math. 75 (2020), Article ID 46, 11 pages. DOI 10.1007/s00025-020-1168-7 | MR 4075300 | Zbl 1439.33007
[8] Guo, V. J. W., Zudilin, W.: Dwork-type supercongruences through a creative $q$-microscope. J. Comb. Theory, Ser. A 178 (2021), Article ID 105362, 37 pages. DOI 10.1016/j.jcta.2020.105362 | MR 4183862 | Zbl 07304668
[9] Li, L., Wang, S.-D.: Proof of a $q$-supercongruence conjectured by Guo and Schlosser. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 114 (2020), Article ID 190, 7 pages. DOI 10.1007/s13398-020-00923-2 | MR 4138479 | Zbl 07258316
[10] Liu, J-C.: 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 | MR 3906342 | Zbl 1423.11015
[11] Liu, J.-C., Petrov, F.: Congruences on sums of $q$-binomial coefficients. Adv. Appl. Math. 116 (2020), Article ID 102003, 11 pages. DOI 10.1016/j.aam.2020.102003 | MR 4056114 | Zbl 07175431
[12] Long, L., Ramakrishna, R.: Some supercongruences occurring in truncated hypergeometric series. Adv. Math. 290 (2016), 773-808. DOI 10.1016/j.aim.2015.11.043 | MR 3451938 | Zbl 1336.33018
[13] Mao, G.-S., Pan, H.: On the divisibility of some truncated hypergeometric series. Acta Arith. 195 (2020), 199-206. DOI 10.4064/aa190511-2-1 | MR 4109895 | Zbl 07221836
[14] Ni, H.-X., Pan, H.: On a conjectured $q$-congruence of Guo and Zeng. Int. J. Number Theory 14 (2018), 1699-1707. DOI 10.1142/S1793042118501038 | MR 3827955 | Zbl 1428.11041
[15] Sun, Z.-W.: On sums of Apéry polynomials and related congruences. J. Number Theory 132 (2012), 2673-2690. DOI 10.1016/j.jnt.2012.05.014 | MR 2954998 | Zbl 1275.11038
[16] Swisher, H.: On the supercongruence conjectures of Van Hamme. Res. Math. Sci. 2 (2015), Article ID 18, 21 pages. DOI 10.1186/s40687-015-0037-6 | MR 3411813 | Zbl 1337.33005
[17] Tauraso, R.: $q$-analogs of some congruences involving Catalan numbers. Adv. Appl. Math. 48 (2012), 603-614. DOI 10.1016/j.aam.2011.12.002 | MR 2920834 | Zbl 1270.11016
[18] Hamme, L. Van: Some conjectures concerning partial sums of generalized hypergeometric series. $p$-Adic Functional Analysis Lecture Notes in Pure and Applied Mathematics 192. Marcel Dekker, New York (1997), 223-236. MR 1459212 | Zbl 0895.11051
[19] Wang, X., Yue, M.: A $q$-analogue of the (A.2) supercongruence of Van Hamme for any prime $p\equiv 3\pmod{4}$. Int. J. Number Theory 16 (2020), 1325-1335. DOI 10.1142/S1793042120500694 | MR 4120479 | Zbl 07219273
[20] Wang, X., Yue, M.: Some $q$-supercongruences from Watson's $_8\phi_7$ transformation formula. Result. Math. 75 (2020), Article ID 71, 15 pages. DOI 10.1007/s00025-020-01195-3 | MR 4091607 | Zbl 1437.33015
[21] Zudilin, W.: Congruences for $q$-binomial coefficients. Ann. Comb. 23 (2019), 1123-1135. DOI 10.1007/s00026-019-00461-8 | MR 4039579 | Zbl 1431.11032

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