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**A common \(q\)-analogue of two supercongruences.**
*(English)*
Zbl 1439.33007

This 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)

### References:

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