A common \(q\)-analogue of two supercongruences.

*(English)*Zbl 1439.33007This paper gives a \(q\)-congruences whose specializations \(q=1\) and \(q= -1\) correspond to supercongrences (B.2) and (H.2) on Van Hammer’s list in \(p\)-adic Functional A analysis. At the end a general common \(q\)-congruence for related hypergeometric sums is given.

This paper also, displays the following historical discussion, taking into account G. Bauer’s formula [J. Reine Angew. Math. 56, 101–121 (1859; ERAM 056.1478cj)],which traditionally gives different methods for proofs of hypergeometric identities, and its special status linked to the fact that it belongs to a family of series for \(\frac{1}{\pi}\) of Ramanujan type and Ramanujan treatment and discussion in [S. Ramanujan, Quart. J. 45, 350–372 (1914; JFM 45.1249.01)].

Different methods used to prove this identity, some used hypergeometric functions, some used other methods such as a creative telescoping method which is a computer proof given by S. B. Ekhad and D. Zeilberger [in: Geometry, analysis and mechanics. Dedicated to Archimedes on his 2281st birthday. Singapore: World Scientific. 107–108 (1994; Zbl 0849.33003)] based on Wilf-Zilberger’s of creative telescoping. The paper also gives some historic links between congruences and their methods of proof.

The authors introduced and executed a new method of creative micro scoping to prove (and reprove) many \(q\)-analogues of classical supercongruences and on \(q\)-congruencies’ goal of this paper is to present new \(q\)-analogue of Van Hammer’s supercongruence, which was given in two theorems.

In section two a family of one parameter \(q\)-congruencies was presented through two theorems and one Lemma. A full discussion including some limiting cases is also presented. At the end one finds a list of references.

This paper also, displays the following historical discussion, taking into account G. Bauer’s formula [J. Reine Angew. Math. 56, 101–121 (1859; ERAM 056.1478cj)],which traditionally gives different methods for proofs of hypergeometric identities, and its special status linked to the fact that it belongs to a family of series for \(\frac{1}{\pi}\) of Ramanujan type and Ramanujan treatment and discussion in [S. Ramanujan, Quart. J. 45, 350–372 (1914; JFM 45.1249.01)].

Different methods used to prove this identity, some used hypergeometric functions, some used other methods such as a creative telescoping method which is a computer proof given by S. B. Ekhad and D. Zeilberger [in: Geometry, analysis and mechanics. Dedicated to Archimedes on his 2281st birthday. Singapore: World Scientific. 107–108 (1994; Zbl 0849.33003)] based on Wilf-Zilberger’s of creative telescoping. The paper also gives some historic links between congruences and their methods of proof.

The authors introduced and executed a new method of creative micro scoping to prove (and reprove) many \(q\)-analogues of classical supercongruences and on \(q\)-congruencies’ goal of this paper is to present new \(q\)-analogue of Van Hammer’s supercongruence, which was given in two theorems.

In section two a family of one parameter \(q\)-congruencies was presented through two theorems and one Lemma. A full discussion including some limiting cases is also presented. At the end one finds a list of references.

Reviewer: Faitori Omer Salem (Tripoli)

##### MSC:

33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |

11A07 | Congruences; primitive roots; residue systems |

11B65 | Binomial coefficients; factorials; \(q\)-identities |

##### Keywords:

basic hypergeometric series; \(q\)-Dixon sum; \(q\)-congruence; supercongruence; creative microscoping
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\textit{V. J. W. Guo} and \textit{W. Zudilin}, Result. Math. 75, No. 2, Paper No. 46, 11 p. (2020; Zbl 1439.33007)

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##### References:

[1] | Bauer, G., Von den coefficienten der Reihen von Kugelfunctionen einer variabeln, J. Reine Angew. Math., 56, 101-121 (1859) |

[2] | Ekhad, Sb; Zeilberger, D.; Wz, A.; Rassias, Jm, Proof of Ramanujan’s formula for \(\pi \), Geometry, Analysis, and Mechanics, 107 (1994), Singapore: World Scientific, Singapore |

[3] | Gasper, G.; Rahman, M., Basic Hypergeometric Series (Encyclopedia of Mathematics and Its Applications 96) (2004), Cambridge: Cambridge University Press, Cambridge |

[4] | Gorodetsky, O., \(q\)-Congruences, with applications to supercongruences and the cyclic sieving phenomenon, Int. J. Number Theory, 15, 1919-1968 (2019) · Zbl 1423.11043 |

[5] | Gu, C-Y; Guo, Vjw, \(q\)-Analogues of two supercongruences of Z.-W. Sun, Czechoslovak Math. J. (2020) |

[6] | Guo, V.J.W.: A further \(q\)-analogue of Van Hamme’s (H.2) supercongruence for primes \(p\equiv 3~(\text{mod} \; 4)\), preprint (2019) |

[7] | Guo, Vjw, A \(q\)-analogue of a Ramanujan-type supercongruence involving central binomial coefficients, J. Math. Anal. Appl., 458, 590-600 (2018) · Zbl 1373.05025 |

[8] | Guo, Vjw, Common \(q\)-analogues of some different supercongruences, Results Math., 74, 131 (2019) · Zbl 1414.33016 |

[9] | Guo, Vjw, \(q\)-Analogues of two “divergent” Ramanujan-type supercongruences, Ramanujan J. (2020) |

[10] | Guo, Vjw; Schlosser, Mj, Some new \(q\)-congruences for truncated basic hypergeometric series: even powers, Results Math., 75, 1 (2020) · Zbl 1442.33007 |

[11] | Guo, V.J.W., Schlosser, M.J.: Some \(q\)-supercongruences from transformation formulas for basic hypergeometric series. Constr. Approx. (to appear) |

[12] | Guo, Vjw; Zeng, J., Some \(q\)-supercongruences for truncated basic hypergeometric series, Acta Arith., 171, 309-326 (2015) · Zbl 1338.11024 |

[13] | Guo, Vjw; Zudilin, W., A \(q\)-microscope for supercongruences, Adv. Math., 346, 329-358 (2019) · Zbl 1464.11028 |

[14] | Guo, Vjw; Zudilin, W., On a \(q\)-deformation of modular forms, J. Math. Anal. Appl., 475, 1636-646 (2019) · Zbl 1445.11014 |

[15] | Liu, J-C, Some supercongruences on truncated \(_3F_2\) hypergeometric series, J. Differ. Equ. Appl., 24, 438-451 (2018) · Zbl 1435.11009 |

[16] | Liu, J-C, On Van Hamme’s (A.2) and (H.2) supercongruences, J. Math. Anal. Appl., 471, 613-622 (2019) · Zbl 1423.11015 |

[17] | Long, L.; Ramakrishna, R., Some supercongruences occurring in truncated hypergeometric series, Adv. Math., 290, 773-808 (2016) · Zbl 1336.33018 |

[18] | Mao, G.-S., Pan, H.: On the divisibility of some truncated hypergeometric series. Acta Arith. (to appear) |

[19] | Mortenson, E., A \(p\)-adic supercongruence conjecture of van Hamme, Proc. Am. Math. Soc., 136, 4321-4328 (2008) · Zbl 1171.11061 |

[20] | Ni, H-X; Pan, H., On a conjectured \(q\)-congruence of Guo and Zeng, Int. J. Number Theory, 14, 1699-1707 (2018) · Zbl 1428.11041 |

[21] | Ramanujan, S., Modular equations and approximations to \(\pi \), Quart. J. Math. Oxf. Ser., 2, 45, 350-372 (1914) · JFM 45.1249.01 |

[22] | Straub, A., Supercongruences for polynomial analogs of the Apéry numbers, Proc. Am. Math. Soc., 147, 1023-1036 (2019) · Zbl 1442.11039 |

[23] | Sun, Z-H, Congruences concerning Legendre polynomials II, J. Number Theory, 133, 1950-1976 (2013) · Zbl 1277.11002 |

[24] | Sun, Z-H, Generalized Legendre polynomials and related supercongruences, J. Number Theory, 143, 293-319 (2014) · Zbl 1353.11005 |

[25] | Sun, Z-W, On sums of Apéry polynomials and related congruences, J. Number Theory, 132, 2673-2690 (2012) · Zbl 1275.11038 |

[26] | Tauraso, R., \(q\)-Analogs of some congruences involving Catalan numbers, Adv. Appl. Math., 48, 603-614 (2009) · Zbl 1270.11016 |

[27] | Van Hamme, L.: Proof of a conjecture of Beukers on Apéry numbers. In: Proceedings of the Conference on \(p\)-Adic Analysis (Houthalen, 1987), pp. 189-195. Vrije Univ. Brussel, Brussels (1986) · Zbl 0634.10004 |

[28] | Van Hamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series. In: \(p\)-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Applied Mathematics, vol. 192, pp. 223-236. Dekker, New York (1997) · Zbl 0895.11051 |

[29] | Zudilin, W., Ramanujan-type supercongruences, J. Number Theory, 129, 1848-1857 (2009) · Zbl 1231.11147 |

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